url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [map_add, hx, smul_eq_mul, hy, mul_add] | case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.map includeRight I
a : R ⊗[A] S
x y x' : R ⊗[A] ↥(Submodule.r... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x✝ : R ⊗[A] S
hx✝ : x✝ ∈ Ideal.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rintro ⟨y, rfl⟩ | case h.mpr
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ (∃ y, (LinearMap.baseChange R (Submodule.restrictScalars A I).subtype) y ... | case h.mpr.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
y : R ⊗[A] ↥(Submodule.restrictScalars A I)
⊢ (LinearMap.baseChange R (Submodule.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x : R ⊗[A] S
⊢ (∃ y, (Linea... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | induction y using TensorProduct.induction_on with
| zero => simp only [_root_.map_zero, Submodule.zero_mem]
| tmul r s =>
rcases s with ⟨s, hs⟩
simp only [restrictScalars_mem] at hs
simp only [baseChange_tmul, coeSubtype]
rw [← mul_one r, ← smul_eq_mul, ← TensorProduct.smul_tmul']
rw [← IsScalarTower.algebraM... | case h.mpr.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
y : R ⊗[A] ↥(Submodule.restrictScalars A I)
⊢ (LinearMap.baseChange R (Submodule.r... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
y : R ⊗[A] ↥(Submodul... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [_root_.map_zero, Submodule.zero_mem] | case h.mpr.intro.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ (LinearMap.baseChange R (Submodule.restrictScalars A I).subtype) 0 ∈ Ideal.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.zero
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ (LinearMap.bas... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rcases s with ⟨s, hs⟩ | case h.mpr.intro.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : ↥(Submodule.restrictScalars A I)
⊢ (LinearMap.baseChange R (Submodu... | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs : s ∈ Submodule.restrictScalars A I
⊢ (LinearMap.baseChange... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : ↥(Subm... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [restrictScalars_mem] at hs | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs : s ∈ Submodule.restrictScalars A I
⊢ (LinearMap.baseChange... | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ (LinearMa... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [baseChange_tmul, coeSubtype] | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ (LinearMa... | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ r ⊗ₜ[A] s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rw [← mul_one r, ← smul_eq_mul, ← TensorProduct.smul_tmul'] | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ r ⊗ₜ[A] s... | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ r • 1 ⊗ₜ[... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | rw [← IsScalarTower.algebraMap_smul (R ⊗[A] S) r, smul_eq_mul] | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ r • 1 ⊗ₜ[... | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ (algebraM... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | apply Ideal.mul_mem_left | case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ (algebraM... | case h.mpr.intro.tmul.mk.a
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ 1 ⊗ₜ[A]... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | exact Ideal.mem_map_of_mem Algebra.TensorProduct.includeRight hs | case h.mpr.intro.tmul.mk.a
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S
hs✝ : s ∈ Submodule.restrictScalars A I
hs : s ∈ I
⊢ 1 ⊗ₜ[A]... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.tmul.mk.a
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
r : R
s : S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | simp only [map_add] | case h.mpr.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x y : R ⊗[A] ↥(Submodule.restrictScalars A I)
hx : (LinearMap.baseChange R (Su... | case h.mpr.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x y : R ⊗[A] ↥(Submodule.restrictScalars A I)
hx : (LinearMap.baseChange R (Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x y : R ⊗[A] ↥(Su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Algebra.TensorProduct.map_includeRight_eq_range_baseChange | [159, 1] | [209, 34] | exact Ideal.add_mem _ hx hy | case h.mpr.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x y : R ⊗[A] ↥(Submodule.restrictScalars A I)
hx : (LinearMap.baseChange R (Su... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.add
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
x y : R ⊗[A] ↥(Su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Ideal.isAugmentation_baseChange | [218, 1] | [228, 32] | unfold Ideal.IsAugmentation | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsAugmentation R (map Alg... | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (Subalgebra.toSub... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Ideal.isAugmentation_baseChange | [218, 1] | [228, 32] | rw [Algebra.baseChange_bot] | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (Subalgebra.toSub... | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (LinearMap.range ... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Ideal.isAugmentation_baseChange | [218, 1] | [228, 32] | rw [Algebra.TensorProduct.map_includeRight_eq_range_baseChange] | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (LinearMap.range ... | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (LinearMap.range ... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/AugmentationIdeal.lean | Ideal.isAugmentation_baseChange | [218, 1] | [228, 32] | exact isCompl_baseChange hI R | R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹ : CommRing R
inst✝ : Algebra A R
⊢ IsCompl (LinearMap.range ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
inst✝⁶ : CommRing R✝
A : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
J : Ideal A
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra A S
I : Ideal S
hI : IsCompl (Subalgebra.toSubmodule ⊥) (Submodule.restrictScalars A I)
R : Type u_4
inst✝¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraComp.lean | algebraMap_injective' | [34, 1] | [38, 100] | rw [IsScalarTower.algebraMap_eq R A K] | R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field A
inst✝⁵ : Algebra R A
inst✝⁴ : IsFractionRing R A
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : Algebra A K
inst✝ : IsScalarTower R A K
⊢ Function.Injective ⇑(algebraMap R K) | R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field A
inst✝⁵ : Algebra R A
inst✝⁴ : IsFractionRing R A
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : Algebra A K
inst✝ : IsScalarTower R A K
⊢ Function.Injective ⇑((algebraMap A K).comp (algebraMap R A)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field A
inst✝⁵ : Algebra R A
inst✝⁴ : IsFractionRing R A
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : Algebra A K
inst✝ : IsScalarTower R A K
⊢ Function.Injective ⇑(algebraMap R K)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/AlgebraComp.lean | algebraMap_injective' | [34, 1] | [38, 100] | apply Function.Injective.comp (RingHom.injective (algebraMap A K)) (IsFractionRing.injective R A) | R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field A
inst✝⁵ : Algebra R A
inst✝⁴ : IsFractionRing R A
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : Algebra A K
inst✝ : IsScalarTower R A K
⊢ Function.Injective ⇑((algebraMap A K).comp (algebraMap R A)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field A
inst✝⁵ : Algebra R A
inst✝⁴ : IsFractionRing R A
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : Algebra A K
inst✝ : IsScalarTower R A K
⊢ Function.Injective ⇑((algebraMap A K).comp (a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | rw [MvPowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto] | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ Filter.Tendsto f u (nhds g) ↔ ∀ (d : ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coeff α d) g)) | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ (∀ (d : Unit →₀ ℕ),
Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g))) ↔
∀ (d : ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coef... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ Filter.Tendsto f u (nhds g) ↔ ∀ (d : ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coeff α d) g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | apply (Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv.forall_congr | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ (∀ (d : Unit →₀ ℕ),
Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g))) ↔
∀ (d : ℕ), Filter.Tendsto (fun i => (coeff α d) (f i)) u (nhds ((coef... | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ ∀ {x : Unit →₀ ℕ},
Filter.Tendsto (fun i => (MvPowerSeries.coeff α x) (f i)) u (nhds ((MvPowerSeries.coeff α x) g)) ↔
Filter.Tendsto (fun i => (coeff α ((Finsupp.LinearEquiv.finsuppUnique ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ (∀ (d : Unit →₀ ℕ),
Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g))) ↔
∀ (d : ℕ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | intro d | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ ∀ {x : Unit →₀ ℕ},
Filter.Tendsto (fun i => (MvPowerSeries.coeff α x) (f i)) u (nhds ((MvPowerSeries.coeff α x) g)) ↔
Filter.Tendsto (fun i => (coeff α ((Finsupp.LinearEquiv.finsuppUnique ℕ... | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) ↔
Filter.Tendsto (fun i => (coeff α ((Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).to... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
⊢ ∀ {x : Unit →₀ ℕ},
Filter.Tendsto (fun i => (MvPowerSeries.coeff α x) (f i)) u (nhds ((MvPowerSeries.coeff α x) g)) ↔
Filter.Ten... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | simp only [LinearEquiv.coe_toEquiv, Finsupp.LinearEquiv.finsuppUnique_apply,
PUnit.default_eq_unit, coeff] | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) ↔
Filter.Tendsto (fun i => (coeff α ((Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).to... | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) ↔
Filter.Tendsto (fun i => (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) ↔
Filter.Tendsto (fun i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | apply iff_of_eq | α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) ↔
Filter.Tendsto (fun i => (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))... | case a
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) =
Filter.Tendsto (fun i => (MvPowerSeries.coeff α (Finsupp.single () (d PUnit... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) ↔
Filter.Tendsto (fun i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | congr | case a
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) =
Filter.Tendsto (fun i => (MvPowerSeries.coeff α (Finsupp.single () (d PUnit... | case a.e_f
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ (fun i => (MvPowerSeries.coeff α d) (f i)) = fun i => (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))) (f i)
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : Topologi... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ Filter.Tendsto (fun i => (MvPowerSeries.coeff α d) (f i)) u (nhds ((MvPowerSeries.coeff α d) g)) =
Filter.Tendsto... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | ext i | case a.e_f
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ (fun i => (MvPowerSeries.coeff α d) (f i)) = fun i => (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))) (f i)
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : Topologi... | case a.e_f.h
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
i : ι
⊢ (MvPowerSeries.coeff α d) (f i) = (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))) (f i)
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_f
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ (fun i => (MvPowerSeries.coeff α d) (f i)) = fun i => (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))) ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | congr | case a.e_f.h
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
i : ι
⊢ (MvPowerSeries.coeff α d) (f i) = (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))) (f i)
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ... | case a.e_f.h.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
i : ι
⊢ d = Finsupp.single () (d PUnit.unit)
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_f.h
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
i : ι
⊢ (MvPowerSeries.coeff α d) (f i) = (MvPowerSeries.coeff α (Finsupp.single () (d PUnit.unit))) (f i)
case ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | all_goals { ext; simp } | case a.e_f.h.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
i : ι
⊢ d = Finsupp.single () (d PUnit.unit)
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_f.h.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
i : ι
⊢ d = Finsupp.single () (d PUnit.unit)
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | ext | case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ d = Finsupp.single () (d PUnit.unit) | case a.e_l₂.e_x.e_a.e_n.h
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ d default = (Finsupp.single () (d PUnit.unit)) default | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_l₂.e_x.e_a.e_n
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ d = Finsupp.single () (d PUnit.unit)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto | [42, 1] | [53, 26] | simp | case a.e_l₂.e_x.e_a.e_n.h
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ d default = (Finsupp.single () (d PUnit.unit)) default | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.e_l₂.e_x.e_a.e_n.h
α : Type u_1
inst✝¹ : TopologicalSpace α
σ : ?m.916
inst✝ : Semiring α
ι : Type u_2
f : ι → α⟦X⟧
u : Filter ι
g : α⟦X⟧
d : Unit →₀ ℕ
⊢ d default = (Finsupp.single () (d PUnit.unit)) default
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.hasSum_of_monomials_self | [149, 1] | [156, 27] | rw [← (Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv.hasSum_iff] | α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
⊢ HasSum (fun d => (monomial α d) ((coeff α d) f)) f | α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
⊢ HasSum ((fun d => (monomial α d) ((coeff α d) f)) ∘ ⇑(Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv) f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
⊢ HasSum (fun d => (monomial α d) ((coeff α d) f)) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.hasSum_of_monomials_self | [149, 1] | [156, 27] | convert MvPowerSeries.hasSum_of_monomials_self f | α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
⊢ HasSum ((fun d => (monomial α d) ((coeff α d) f)) ∘ ⇑(Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv) f | case h.e'_5.h.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ ((fun d => (monomial α d) ((coeff α d) f)) ∘ ⇑(Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv) x✝ =
(MvPowerSeries.monomial α x✝) ((MvPowerSeries.coeff α x✝) f) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
⊢ HasSum ((fun d => (monomial α d) ((coeff α d) f)) ∘ ⇑(Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.hasSum_of_monomials_self | [149, 1] | [156, 27] | simp only [LinearEquiv.coe_toEquiv, comp_apply, monomial, coeff,
Finsupp.LinearEquiv.finsuppUnique_apply, PUnit.default_eq_unit] | case h.e'_5.h.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ ((fun d => (monomial α d) ((coeff α d) f)) ∘ ⇑(Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv) x✝ =
(MvPowerSeries.monomial α x✝) ((MvPowerSeries.coeff α x✝) f) | case h.e'_5.h.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ (MvPowerSeries.monomial α (Finsupp.single () (x✝ PUnit.unit)))
((MvPowerSeries.coeff α (Finsupp.single () (x✝ PUnit.unit))) f) =
(MvPowerSeries.monomial α x✝) ((MvPowerSerie... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ ((fun d => (monomial α d) ((coeff α d) f)) ∘ ⇑(Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv) x✝ =
(MvPowerSerie... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.hasSum_of_monomials_self | [149, 1] | [156, 27] | congr | case h.e'_5.h.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ (MvPowerSeries.monomial α (Finsupp.single () (x✝ PUnit.unit)))
((MvPowerSeries.coeff α (Finsupp.single () (x✝ PUnit.unit))) f) =
(MvPowerSeries.monomial α x✝) ((MvPowerSerie... | case h.e'_5.h.h.h.e_5.h.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () (x✝ PUnit.unit) = x✝
case h.e'_5.h.h.h.e_6.h.e_a.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Un... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ (MvPowerSeries.monomial α (Finsupp.single () (x✝ PUnit.unit)))
((MvPowerSeries.coeff α (Finsupp.single () (x✝ PUnit.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.hasSum_of_monomials_self | [149, 1] | [156, 27] | all_goals { ext ; simp } | case h.e'_5.h.h.h.e_5.h.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () (x✝ PUnit.unit) = x✝
case h.e'_5.h.h.h.e_6.h.e_a.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Un... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.h.h.e_5.h.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () (x✝ PUnit.unit) = x✝
case h.e'_5.h.h.h.e_6.h.e_a.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.hasSum_of_monomials_self | [149, 1] | [156, 27] | ext | case h.e'_5.h.h.h.e_6.h.e_a.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () (x✝ PUnit.unit) = x✝ | case h.e'_5.h.h.h.e_6.h.e_a.e_n.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ (Finsupp.single () (x✝ PUnit.unit)) default = x✝ default | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.h.h.e_6.h.e_a.e_n
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () (x✝ PUnit.unit) = x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/PowerSeries/Topology.lean | PowerSeries.hasSum_of_monomials_self | [149, 1] | [156, 27] | simp | case h.e'_5.h.h.h.e_6.h.e_a.e_n.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ (Finsupp.single () (x✝ PUnit.unit)) default = x✝ default | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.h.h.e_6.h.e_a.e_n.h
α : Type u_1
inst✝¹ : Semiring α
inst✝ : TopologicalSpace α
f : α⟦X⟧
e_1✝ : α⟦X⟧ = MvPowerSeries Unit α
x✝ : Unit →₀ ℕ
⊢ (Finsupp.single () (x✝ PUnit.unit)) default = x✝ default
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | suffices (rTensor_fgEquiv R M N).toLinearMap.comp
(Module.DirectLimit.of R (Submodules_fg R M) (fun P ↦ P.val ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules_fg_inclusion R M P Q hPQ)) P)
= LinearMap.rTensor N (Submodule.subtype P.val) by
exact DFunLike.congr_fun this u | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
⊢ (rTensor_fgEquiv R M N)
((Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.r... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
⊢ ↑(rTensor_fgEquiv R M N) ∘ₗ
Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.r... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
⊢ (rTensor_fgEquiv R M N)
((Module.DirectLimit.of R (Submodules_fg R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | ext p n | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
⊢ ↑(rTensor_fgEquiv R M N) ∘ₗ
Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.r... | case a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry
(↑(rTensor_fgEquiv R M N) ∘ₗ
Module.DirectLimit.of R (Submod... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
⊢ ↑(rTensor_fgEquiv R M N) ∘ₗ
Module.DirectLimit.of R (Submodules_fg R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | simp only [rTensor_fgEquiv, AlgebraTensorModule.curry_apply, curry_apply,
LinearMap.coe_restrictScalars, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
LinearEquiv.trans_apply, directLimitLeft_symm_of_tmul, LinearEquiv.rTensor_tmul,
LinearMap.rTensor_tmul, Submodule.coeSubtype] | case a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry
(↑(rTensor_fgEquiv R M N) ∘ₗ
Module.DirectLimit.of R (Submod... | case a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_equiv R M)
((Module.DirectLimit.of R (Submodules_fg R M) (fun i => ↥↑i)
(fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry
(↑(rTensor_... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | congr | case a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_equiv R M)
((Module.DirectLimit.of R (Submodules_fg R M) (fun i => ↥↑i)
(fu... | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_equiv R M)
((Module.DirectLimit.of R (Submodules_fg R M) (fun i => ↥↑i) (fun P Q hPQ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_equiv R M)
((Module.Direct... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | simp only [Submodules_fg_equiv, LinearEquiv.ofBijective_apply] | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_equiv R M)
((Module.DirectLimit.of R (Submodules_fg R M) (fun i => ↥↑i) (fun P Q hPQ ... | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_map R M)
((Module.DirectLimit.of R (Submodules_fg R M) (fun i => ↥↑i) (fun P Q hPQ =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_equiv R M)
((Module.Dire... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | simp only [Submodules_fg_map] | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_map R M)
((Module.DirectLimit.of R (Submodules_fg R M) (fun i => ↥↑i) (fun P Q hPQ =>... | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Module.DirectLimit.lift R (Submodules_fg R M) (fun P => ↥↑P) (Submodules_fg_inclusion R M) (fun P => (↑P)... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Submodules_fg_map R M)
((Module.Direct... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | simp only [Module.DirectLimit.lift_of] | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Module.DirectLimit.lift R (Submodules_fg R M) (fun P => ↥↑P) (Submodules_fg_inclusion R M) (fun P => (↑P)... | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (↑P).subtype p = ↑p | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (Module.DirectLimit.lift R (Submodules_fg R M... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | simp only [Submodule.coeSubtype] | case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (↑P).subtype p = ↑p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h.e_m
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
p : ↥↑P
n : N
⊢ (↑P).subtype p = ↑p
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fgEquiv_of | [126, 1] | [145, 35] | exact DFunLike.congr_fun this u | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
this :
↑(rTensor_fgEquiv R M N) ∘ₗ
Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => Line... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
this :
↑(rTensor_fgEquiv R M N) ∘ₗ
Module.DirectLimit.of R (Submodul... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | rw [← LinearMap.id_apply (R := R) p] | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP)) p = p | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP)) (LinearMap.id p) = LinearMap.id p | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP)) p = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | apply DFunLike.congr_fun | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP)) (LinearMap.id p) = LinearMap.id p | case h₁
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP) = LinearMap.id | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP)) (Lin... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | ext p n | case h₁
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP) = LinearMap.id | case h₁.a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p✝ : ↥↑P ⊗[R] N
hP : P ≤ P
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry (LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP))) p) n =
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p : ↥↑P ⊗[R] N
hP : P ≤ P
⊢ LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | rfl | case h₁.a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p✝ : ↥↑P ⊗[R] N
hP : P ≤ P
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry (LinearMap.rTensor N (Submodules_fg_inclusion R M P P hP))) p) n =
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.a.h.h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodules_fg R M
p✝ : ↥↑P ⊗[R] N
hP : P ≤ P
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry (Linea... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | rw [← LinearMap.comp_apply, ← LinearMap.rTensor_comp] | R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R✝ M Q R hRQ))
((LinearMap.rTensor N (Submodules_... | R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R✝ M Q R hRQ ∘ₗ Submodules_fg_inclusion R✝ M P Q hPQ)) ... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ (LinearMap.rTensor N (Submodules_fg_incl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | apply DFunLike.congr_fun | R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ (LinearMap.rTensor N (Submodules_fg_inclusion R✝ M Q R hRQ ∘ₗ Submodules_fg_inclusion R✝ M P Q hPQ)) ... | case h₁
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ LinearMap.rTensor N (Submodules_fg_inclusion R✝ M Q R hRQ ∘ₗ Submodules_fg_inclusion R✝ M P Q... | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ (LinearMap.rTensor N (Submodules_fg_incl... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | ext p n | case h₁
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ LinearMap.rTensor N (Submodules_fg_inclusion R✝ M Q R hRQ ∘ₗ Submodules_fg_inclusion R✝ M P Q... | case h₁.a.h.h
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p✝ : ↥↑P ⊗[R✝] N
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry
(LinearMap.rTensor N (Submodules_f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p : ↥↑P ⊗[R✝] N
⊢ LinearMap.rTensor N (Submodules_... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | simp | case h₁.a.h.h
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p✝ : ↥↑P ⊗[R✝] N
p : ↥↑P
n : N
⊢ ((AlgebraTensorModule.curry
(LinearMap.rTensor N (Submodules_f... | case h₁.a.h.h
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p✝ : ↥↑P ⊗[R✝] N
p : ↥↑P
n : N
⊢ (Submodules_fg_inclusion R✝ M Q R hRQ) ((Submodules_fg_inclusion R✝ M P ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.a.h.h
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p✝ : ↥↑P ⊗[R✝] N
p : ↥↑P
n : N
⊢ ((AlgebraTe... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | rTensor_fg_directedSystem | [157, 1] | [170, 10] | rfl | case h₁.a.h.h
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p✝ : ↥↑P ⊗[R✝] N
p : ↥↑P
n : N
⊢ (Submodules_fg_inclusion R✝ M Q R hRQ) ((Submodules_fg_inclusion R✝ M P ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.a.h.h
R✝ : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R✝
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
inst✝¹ : AddCommGroup N
inst✝ : Module R✝ N
P Q R : Submodules_fg R✝ M
hPQ : P ≤ Q
hRQ : Q ≤ R
p✝ : ↥↑P ⊗[R✝] N
p : ↥↑P
n : N
⊢ (Submodules... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.exists_of_fg | [174, 1] | [178, 62] | let ⟨P, u, hu⟩ := Module.DirectLimit.exists_of ((rTensor_fgEquiv R M N).symm t) | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∃ P, P.FG ∧ t ∈ LinearMap.range (LinearMap.rTensor N P.subtype) | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
hu :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submo... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∃ P, P.FG ∧ t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.exists_of_fg | [174, 1] | [178, 62] | use P.val, P.property, u | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
hu :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submo... | case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
hu :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
hu :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.exists_of_fg | [174, 1] | [178, 62] | rw [← rTensor_fgEquiv_of, hu, LinearEquiv.apply_symm_apply] | case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
hu :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : M ⊗[R] N
P : Submodules_fg R M
u : ↥↑P ⊗[R] N
hu :
(Module.DirectLimit.of R (Submodules_fg R M) (fun... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq | [180, 1] | [192, 44] | rw [← sub_eq_zero, ← map_sub, ← rTensor_fgEquiv_of R M N ⟨P,hP⟩,
LinearEquiv.map_eq_zero_iff] at h | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P.subtype) t'
⊢ ∃ Q,
∃ (hPQ : P ≤ Q),
Q.FG ∧ (LinearMap.rT... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules_... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P.subty... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq | [180, 1] | [192, 44] | have := rTensor_fg_directedSystem R M N | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules_... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules_... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq | [180, 1] | [192, 44] | have := Module.DirectLimit.of.zero_exact h | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules_... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules_... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq | [180, 1] | [192, 44] | let ⟨Q, hPQ, h⟩ := this | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules_... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h✝ :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq | [180, 1] | [192, 44] | use Q.val, Subtype.coe_le_coe.mpr hPQ, Q.property | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h✝ :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N (Submodules... | case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h✝ :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h✝ :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq | [180, 1] | [192, 44] | simpa only [sub_eq_zero, map_sub] using h | case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h✝ :
(Module.DirectLimit.of R (Submodules_fg R M) (fun P => ↥↑P ⊗[R] N)
(fun P Q hPQ => LinearMap.rTensor N ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t t' : ↥P ⊗[R] N
h✝ :
(Module.DirectLimit.of R (Submodules_fg R M) (fun ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | let P'' := P ⊔ P' | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t'
⊢ ∃ Q,
... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t'
P'' : Sub... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | let _hP_le := (le_sup_left : P ≤ P'') | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t'
P'' : Sub... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t'
P'' : Sub... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | let _hP'_le := (le_sup_right : P' ≤ P'') | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t'
P'' : Sub... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t'
P'' : Sub... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | rw [← Submodule.subtype_comp_inclusion _ _ _hP_le] at h | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t'
P'' : Sub... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
h : (LinearMap.rTens... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
h : (LinearMap.rTensor N... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | rw [← Submodule.subtype_comp_inclusion _ _ _hP'_le] at h | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
h : (LinearMap.rTens... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le : P' ≤ P ⊔ P... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | simp only [LinearMap.rTensor_comp, LinearMap.coe_comp, Function.comp_apply] at h | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le : P' ≤ P ⊔ P... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le : P' ≤ P ⊔ P... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | let ⟨Q, hQ_le, hQ, h⟩ := TensorProduct.eq_of_fg_of_subtype_eq
(Submodule.FG.sup hP hP') _ _ h | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le : P' ≤ P ⊔ P... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le : P' ≤ P ⊔ P... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | use Q, le_trans _hP_le hQ_le, le_trans _hP'_le hQ_le, hQ | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le : P' ≤ P ⊔ P... | case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | simp only [← LinearMap.comp_apply, ← LinearMap.rTensor_comp, Submodule.subtype_comp_inclusion] at h | case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le :... | case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.eq_of_fg_of_subtype_eq' | [194, 1] | [211, 10] | exact h | case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodule R M := P ⊔ P'
_hP_le : P ≤ P ⊔ P' := le_sup_left
_hP'_le :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : Submodule R M
hP : P.FG
t : ↥P ⊗[R] N
P' : Submodule R M
hP' : P'.FG
t' : ↥P' ⊗[R] N
P'' : Submodu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | let ⟨P, hP, ht⟩ := TensorProduct.exists_of_fg t | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
⊢ ∃ A, A.FG ∧ t ∈ LinearMap.range (LinearMap.rTensor N A.val.toLinearMap) | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
hP : P.FG
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
⊢ ∃ A, A.FG ∧ t ∈ LinearMap.range (LinearMap.rTensor N A.val.toLinearMap) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
⊢ ∃ A, A.FG ∧ t ∈ LinearMap.range (LinearMap.rTensor N A.val.toLinearMap)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | obtain ⟨s, hs⟩ := hP | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
hP : P.FG
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
⊢ ∃ A, A.FG ∧ t ∈ LinearMap.range (LinearMap.rTensor N A.val.toLinearMap) | case intro
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
⊢ ∃ A, A.FG ∧ t ∈ LinearMap.range (L... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
hP : P.FG
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
⊢ ∃ A, A.FG ∧ t ∈ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | use Algebra.adjoin R s, Subalgebra.fg_adjoin_finset _ | case intro
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
⊢ ∃ A, A.FG ∧ t ∈ LinearMap.range (L... | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
⊢ t ∈ LinearMap.range (LinearMap.rTe... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | have : P ≤ Subalgebra.toSubmodule (Algebra.adjoin R (s : Set S)) := by
simp only [← hs, Submodule.span_le, Subalgebra.coe_toSubmodule]
exact Algebra.subset_adjoin | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
⊢ t ∈ LinearMap.range (LinearMap.rTe... | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
this : P ≤ Subalgebra.toSubmodule (A... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | rw [← Submodule.subtype_comp_inclusion P _ this,
LinearMap.rTensor_comp] at ht | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
this : P ≤ Subalgebra.toSubmodule (A... | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
s : Finset S
hs : Submodule.span R ↑s = P
this : P ≤ Subalgebra.toSubmodule (Algebra.adjoin R ↑s)
ht :
t ∈
LinearMap.range
... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | exact LinearMap.range_comp_le_range _ _ ht | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
s : Finset S
hs : Submodule.span R ↑s = P
this : P ≤ Subalgebra.toSubmodule (Algebra.adjoin R ↑s)
ht :
t ∈
LinearMap.range
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
s : Finset S
hs : Submodule.span R ↑s = P
this : P ≤ Subalgebra.toSubmodul... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | simp only [← hs, Submodule.span_le, Subalgebra.coe_toSubmodule] | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
⊢ P ≤ Subalgebra.toSubmodule (Algebra.adjoin R ... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
⊢ ↑s ⊆ ↑(Algebra.adjoin R ↑s) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.exists_of_fg | [220, 1] | [231, 45] | exact Algebra.subset_adjoin | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.span R ↑s = P
⊢ ↑s ⊆ ↑(Algebra.adjoin R ↑s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
t : S ⊗[R] N
P : Submodule R S
ht : t ∈ LinearMap.range (LinearMap.rTensor N P.subtype)
s : Finset S
hs : Submodule.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | classical
let ⟨P, hP, ⟨u, hu⟩⟩ := TensorProduct.exists_of_fg t
let ⟨P', hP', ⟨u', hu'⟩⟩ := TensorProduct.exists_of_fg t'
let P₁ := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
have hP₁ : Submodule.FG P₁ := Submodule.FG.map _ (Submodule.FG.sup hP hP')
let j : P →ₗ[R] P₁ := LinearMap.restrict (Subalgebra.toS... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
⊢ ∃ B,
∃ (hAB : A ≤ B),
B.FG ∧... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let ⟨P, hP, ⟨u, hu⟩⟩ := TensorProduct.exists_of_fg t | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
⊢ ∃ B,
∃ (hAB : A ≤ B),
B.FG ∧... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let ⟨P', hP', ⟨u', hu'⟩⟩ := TensorProduct.exists_of_fg t' | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let P₁ := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P') | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | have hP₁ : Submodule.FG P₁ := Submodule.FG.map _ (Submodule.FG.sup hP hP') | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let j : P →ₗ[R] P₁ := LinearMap.restrict (Subalgebra.toSubmodule A).subtype
(fun p hp ↦ by
simp only [Submodule.coeSubtype, Submodule.map_sup, P₁]
apply Submodule.mem_sup_left
use p; simp only [SetLike.mem_coe]; exact ⟨hp, rfl⟩) | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let j' : P' →ₗ[R] P₁ := LinearMap.restrict (Subalgebra.toSubmodule A).subtype
(fun p hp ↦ by
simp only [Submodule.coeSubtype, Submodule.map_sup, P₁]
apply Submodule.mem_sup_right
use p; simp only [SetLike.mem_coe]; exact ⟨hp, rfl⟩) | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | set u₁ := LinearMap.rTensor N j u with hu₁ | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | set u'₁ := LinearMap.rTensor N j' u' with hu'₁ | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | have : LinearMap.rTensor N P₁.subtype u₁ = LinearMap.rTensor N P₁.subtype u'₁ := by
rw [hu₁, hu'₁]
simp only [← LinearMap.comp_apply, ← LinearMap.rTensor_comp]
have hj₁ : P₁.subtype ∘ₗ j = (Subalgebra.val A).toLinearMap ∘ₗ P.subtype := by ext; rfl
have hj'₁ : P₁.subtype ∘ₗ j' = (Subalgebra.val A).toLinearMap ∘ₗ... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let ⟨P'₁, hP₁_le, hP'₁, h⟩ := TensorProduct.eq_of_fg_of_subtype_eq hP₁ _ _ this | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R]... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let ⟨s, hs⟩ := hP'₁ | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let ⟨w, hw⟩ := hA | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | let B := Algebra.adjoin R ((s ∪ w : Finset S) : Set S) | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | have hBA : A ≤ B := by
simp only [B, ← hw]
apply Algebra.adjoin_mono
simp only [Finset.coe_union, Set.subset_union_right] | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | use B, hBA, Subalgebra.fg_adjoin_finset _ | R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R... | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | rw [← hu, ← hu'] | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG... | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/DirectLimit.lean | TensorProduct.Algebra.eq_of_fg_of_subtype_eq | [233, 1] | [288, 24] | simp only [← LinearMap.comp_apply, ← LinearMap.rTensor_comp] | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG... | case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap... |
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