statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
sum_add_distrib' {ι} (f g : ι → cardinal) :
cardinal.sum (λ i, f i + g i) = sum f + sum g | sum_add_distrib f g | theorem | cardinal.sum_add_distrib' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"cardinal.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_sum {ι : Type u} (f : ι → cardinal.{v}) :
cardinal.lift.{w} (cardinal.sum f) = cardinal.sum (λ i, cardinal.lift.{w} (f i)) | equiv.cardinal_eq $ equiv.ulift.trans $ equiv.sigma_congr_right $ λ a, nonempty.some $
by rw [←lift_mk_eq, mk_out, mk_out, lift_lift] | theorem | cardinal.lift_sum | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.sum",
"equiv.sigma_congr_right",
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_sum {ι} (f g : ι → cardinal) (H : ∀ i, f i ≤ g i) : sum f ≤ sum g | ⟨(embedding.refl _).sigma_map $ λ i, classical.choice $
by have := H i; rwa [← quot.out_eq (f i), ← quot.out_eq (g i)] at this⟩ | theorem | cardinal.sum_le_sum | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"quot.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_mk_mul_of_mk_preimage_le {c : cardinal} (f : α → β) (hf : ∀ b : β, #(f ⁻¹' {b}) ≤ c) :
#α ≤ #β * c | by simpa only [←mk_congr (@equiv.sigma_fiber_equiv α β f), mk_sigma, ←sum_const']
using sum_le_sum _ _ hf | lemma | cardinal.mk_le_mk_mul_of_mk_preimage_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.sigma_fiber_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v} {c : cardinal}
(f : α → β) (hf : ∀ b : β, lift.{v} #(f ⁻¹' {b}) ≤ c) :
lift.{v} #α ≤ lift.{u} #β * c | mk_le_mk_mul_of_mk_preimage_le (λ x : ulift.{v} α, ulift.up.{u} (f x.1)) $ ulift.forall.2 $ λ b,
(mk_congr $ (equiv.ulift.image _).trans (equiv.trans
(by { rw [equiv.image_eq_preimage], simp [set.preimage] }) equiv.ulift.symm)).trans_le (hf b) | lemma | cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.image_eq_preimage",
"equiv.trans",
"set.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_range {ι : Type u} (f : ι → cardinal.{max u v}) : bdd_above (set.range f) | ⟨_, by { rintros a ⟨i, rfl⟩, exact le_sum f i }⟩ | theorem | cardinal.bdd_above_range | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"set.range"
] | The range of an indexed cardinal function, whose outputs live in a higher universe than the
inputs, is always bounded above. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_above_iff_small {s : set cardinal.{u}} : bdd_above s ↔ small.{u} s | ⟨λ ⟨a, ha⟩, @small_subset _ (Iic a) s (λ x h, ha h) _, begin
rintro ⟨ι, ⟨e⟩⟩,
suffices : range (λ x : ι, (e.symm x).1) = s,
{ rw ←this,
apply bdd_above_range.{u u} },
ext x,
refine ⟨_, λ hx, ⟨e ⟨x, hx⟩, _⟩⟩,
{ rintro ⟨a, rfl⟩,
exact (e.symm a).prop },
{ simp_rw [subtype.val_eq_coe, equiv.symm_appl... | theorem | cardinal.bdd_above_iff_small | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"equiv.symm_apply_apply",
"small_subset",
"subtype.val_eq_coe"
] | A set of cardinals is bounded above iff it's small, i.e. it corresponds to an usual ZFC set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_above_of_small (s : set cardinal.{u}) [h : small.{u} s] : bdd_above s | bdd_above_iff_small.2 h | theorem | cardinal.bdd_above_of_small | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_image (f : cardinal.{u} → cardinal.{max u v}) {s : set cardinal.{u}}
(hs : bdd_above s) : bdd_above (f '' s) | by { rw bdd_above_iff_small at hs ⊢, exactI small_lift _ } | theorem | cardinal.bdd_above_image | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"small_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_range_comp {ι : Type u} {f : ι → cardinal.{v}} (hf : bdd_above (range f))
(g : cardinal.{v} → cardinal.{max v w}) : bdd_above (range (g ∘ f)) | by { rw range_comp, exact bdd_above_image g hf } | theorem | cardinal.bdd_above_range_comp | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_le_sum {ι} (f : ι → cardinal) : supr f ≤ sum f | csupr_le' $ le_sum _ | theorem | cardinal.supr_le_sum | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"csupr_le'",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_supr_lift {ι : Type u} (f : ι → cardinal.{max u v}) :
sum f ≤ (#ι).lift * supr f | begin
rw [←(supr f).lift_id, ←lift_umax, lift_umax.{(max u v) u}, ←sum_const],
exact sum_le_sum _ _ (le_csupr $ bdd_above_range.{u v} f)
end | theorem | cardinal.sum_le_supr_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"le_csupr",
"lift",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_supr {ι : Type u} (f : ι → cardinal.{u}) : sum f ≤ #ι * supr f | by { rw ←lift_id (#ι), exact sum_le_supr_lift f } | theorem | cardinal.sum_le_supr | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_nat_eq_add_sum_succ (f : ℕ → cardinal.{u}) :
cardinal.sum f = f 0 + cardinal.sum (λ i, f (i + 1)) | begin
refine (equiv.sigma_nat_succ (λ i, quotient.out (f i))).cardinal_eq.trans _,
simp only [mk_sum, mk_out, lift_id, mk_sigma],
end | theorem | cardinal.sum_nat_eq_add_sum_succ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.sum",
"equiv.sigma_nat_succ",
"quotient.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_of_empty {ι} (f : ι → cardinal) [is_empty ι] : supr f = 0 | csupr_of_empty f | theorem | cardinal.supr_of_empty | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"csupr_of_empty",
"is_empty",
"supr",
"supr_of_empty"
] | A variant of `csupr_of_empty` but with `0` on the RHS for convenience | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk_shrink (α : Type u) [small.{v} α] :
cardinal.lift.{max u w} (# (shrink.{v} α)) = cardinal.lift.{max v w} (# α) | lift_mk_eq.2 ⟨(equiv_shrink α).symm⟩ | lemma | cardinal.lift_mk_shrink | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv_shrink"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mk_shrink' (α : Type u) [small.{v} α] :
cardinal.lift.{u} (# (shrink.{v} α)) = cardinal.lift.{v} (# α) | lift_mk_shrink.{u v 0} α | lemma | cardinal.lift_mk_shrink' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mk_shrink'' (α : Type (max u v)) [small.{v} α] :
cardinal.lift.{u} (# (shrink.{v} α)) = # α | by rw [← lift_umax', lift_mk_shrink.{(max u v) v 0} α, ← lift_umax, lift_id] | lemma | cardinal.lift_mk_shrink'' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod {ι : Type u} (f : ι → cardinal) : cardinal | #(Π i, (f i).out) | def | cardinal.prod | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | The indexed product of cardinals is the cardinality of the Pi type
(dependent product). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pi {ι : Type u} (α : ι → Type v) : #(Π i, α i) = prod (λ i, #(α i)) | mk_congr $ equiv.Pi_congr_right $ λ i, out_mk_equiv.symm | theorem | cardinal.mk_pi | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.Pi_congr_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_const (ι : Type u) (a : cardinal.{v}) :
prod (λ i : ι, a) = lift.{u} a ^ lift.{v} (#ι) | induction_on a $ λ α, mk_congr $ equiv.Pi_congr equiv.ulift.symm $
λ i, out_mk_equiv.trans equiv.ulift.symm | theorem | cardinal.prod_const | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.Pi_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_const' (ι : Type u) (a : cardinal.{u}) : prod (λ _:ι, a) = a ^ #ι | induction_on a $ λ α, (mk_pi _).symm | theorem | cardinal.prod_const' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod {ι} (f g : ι → cardinal) (H : ∀ i, f i ≤ g i) : prod f ≤ prod g | ⟨embedding.Pi_congr_right $ λ i, classical.choice $
by have := H i; rwa [← mk_out (f i), ← mk_out (g i)] at this⟩ | theorem | cardinal.prod_le_prod | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_zero {ι} (f : ι → cardinal.{u}) : prod f = 0 ↔ ∃ i, f i = 0 | by { lift f to ι → Type u using λ _, trivial, simp only [mk_eq_zero_iff, ← mk_pi, is_empty_pi] } | theorem | cardinal.prod_eq_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"is_empty_pi",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ne_zero {ι} (f : ι → cardinal) : prod f ≠ 0 ↔ ∀ i, f i ≠ 0 | by simp [prod_eq_zero] | theorem | cardinal.prod_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_prod {ι : Type u} (c : ι → cardinal.{v}) :
lift.{w} (prod c) = prod (λ i, lift.{w} (c i)) | begin
lift c to ι → Type v using λ _, trivial,
simp only [← mk_pi, ← mk_ulift],
exact mk_congr (equiv.ulift.trans $ equiv.Pi_congr_right $ λ i, equiv.ulift.symm)
end | theorem | cardinal.lift_prod | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.Pi_congr_right",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_of_fintype {α : Type u} [fintype α] (f : α → cardinal.{v}) :
prod f = cardinal.lift.{u} (∏ i, f i) | begin
revert f,
refine fintype.induction_empty_option _ _ _ α,
{ introsI α β hβ e h f,
letI := fintype.of_equiv β e.symm,
rw [←e.prod_comp f, ←h],
exact mk_congr (e.Pi_congr_left _).symm },
{ intro f,
rw [fintype.univ_pempty, finset.prod_empty, lift_one, cardinal.prod, mk_eq_one] },
{ intros α... | lemma | cardinal.prod_eq_of_fintype | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.prod",
"equiv.pi_option_equiv_prod",
"finset.prod_empty",
"fintype",
"fintype.induction_empty_option",
"fintype.of_equiv",
"fintype.prod_option",
"fintype.univ_pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_Inf (s : set cardinal) : lift (Inf s) = Inf (lift '' s) | begin
rcases eq_empty_or_nonempty s with rfl | hs,
{ simp },
{ exact lift_monotone.map_Inf hs }
end | theorem | cardinal.lift_Inf | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_infi {ι} (f : ι → cardinal) : lift (infi f) = ⨅ i, lift (f i) | by { unfold infi, convert lift_Inf (range f), rw range_comp } | theorem | cardinal.lift_infi | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"infi",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_down {a : cardinal.{u}} {b : cardinal.{max u v}} :
b ≤ lift a → ∃ a', lift a' = b | induction_on₂ a b $ λ α β,
by rw [← lift_id (#β), ← lift_umax, ← lift_umax.{u v}, lift_mk_le]; exact
λ ⟨f⟩, ⟨#(set.range f), eq.symm $ lift_mk_eq.2
⟨embedding.equiv_of_surjective
(embedding.cod_restrict _ f set.mem_range_self)
$ λ ⟨a, ⟨b, e⟩⟩, ⟨b, subtype.eq e⟩⟩⟩ | theorem | cardinal.lift_down | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"embedding.cod_restrict",
"lift",
"set.mem_range_self",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_lift_iff {a : cardinal.{u}} {b : cardinal.{max u v}} :
b ≤ lift a ↔ ∃ a', lift a' = b ∧ a' ≤ a | ⟨λ h, let ⟨a', e⟩ := lift_down h in ⟨a', e, lift_le.1 $ e.symm ▸ h⟩,
λ ⟨a', e, h⟩, e ▸ lift_le.2 h⟩ | theorem | cardinal.le_lift_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_lift_iff {a : cardinal.{u}} {b : cardinal.{max u v}} :
b < lift a ↔ ∃ a', lift a' = b ∧ a' < a | ⟨λ h, let ⟨a', e⟩ := lift_down h.le in ⟨a', e, lift_lt.1 $ e.symm ▸ h⟩,
λ ⟨a', e, h⟩, e ▸ lift_lt.2 h⟩ | theorem | cardinal.lt_lift_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_succ (a) : lift (succ a) = succ (lift a) | le_antisymm
(le_of_not_gt $ λ h, begin
rcases lt_lift_iff.1 h with ⟨b, e, h⟩,
rw [lt_succ_iff, ← lift_le, e] at h,
exact h.not_lt (lt_succ _)
end)
(succ_le_of_lt $ lift_lt.2 $ lt_succ a) | theorem | cardinal.lift_succ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_umax_eq {a : cardinal.{u}} {b : cardinal.{v}} :
lift.{max v w} a = lift.{max u w} b ↔ lift.{v} a = lift.{u} b | by rw [←lift_lift, ←lift_lift, lift_inj] | theorem | cardinal.lift_umax_eq | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_min {a b : cardinal} : lift (min a b) = min (lift a) (lift b) | lift_monotone.map_min | theorem | cardinal.lift_min | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_max {a b : cardinal} : lift (max a b) = max (lift a) (lift b) | lift_monotone.map_max | theorem | cardinal.lift_max | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_Sup {s : set cardinal} (hs : bdd_above s) : lift.{u} (Sup s) = Sup (lift.{u} '' s) | begin
apply ((le_cSup_iff' (bdd_above_image _ hs)).2 (λ c hc, _)).antisymm (cSup_le' _),
{ by_contra h,
obtain ⟨d, rfl⟩ := cardinal.lift_down (not_le.1 h).le,
simp_rw lift_le at h hc,
rw cSup_le_iff' hs at h,
exact h (λ a ha, lift_le.1 $ hc (mem_image_of_mem _ ha)) },
{ rintros i ⟨j, hj, rfl⟩,
... | lemma | cardinal.lift_Sup | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"by_contra",
"cSup_le'",
"cSup_le_iff'",
"cardinal",
"cardinal.lift_down",
"le_cSup",
"le_cSup_iff'"
] | The lift of a supremum is the supremum of the lifts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_supr {ι : Type v} {f : ι → cardinal.{w}} (hf : bdd_above (range f)) :
lift.{u} (supr f) = ⨆ i, lift.{u} (f i) | by rw [supr, supr, lift_Sup hf, ←range_comp] | lemma | cardinal.lift_supr | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"supr"
] | The lift of a supremum is the supremum of the lifts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_supr_le {ι : Type v} {f : ι → cardinal.{w}} {t : cardinal} (hf : bdd_above (range f))
(w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (supr f) ≤ t | by { rw lift_supr hf, exact csupr_le' w } | lemma | cardinal.lift_supr_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"cardinal",
"csupr_le'",
"supr"
] | To prove that the lift of a supremum is bounded by some cardinal `t`,
it suffices to show that the lift of each cardinal is bounded by `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_supr_le_iff {ι : Type v} {f : ι → cardinal.{w}} (hf : bdd_above (range f))
{t : cardinal} : lift.{u} (supr f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t | by { rw lift_supr hf, exact csupr_le_iff' (bdd_above_range_comp hf _) } | lemma | cardinal.lift_supr_le_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"cardinal",
"csupr_le_iff'",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_supr_le_lift_supr
{ι : Type v} {ι' : Type v'} {f : ι → cardinal.{w}} {f' : ι' → cardinal.{w'}}
(hf : bdd_above (range f)) (hf' : bdd_above (range f'))
{g : ι → ι'} (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) :
lift.{w'} (supr f) ≤ lift.{w} (supr f') | begin
rw [lift_supr hf, lift_supr hf'],
exact csupr_mono' (bdd_above_range_comp hf' _) (λ i, ⟨_, h i⟩)
end | lemma | cardinal.lift_supr_le_lift_supr | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"csupr_mono'",
"supr"
] | To prove an inequality between the lifts to a common universe of two different supremums,
it suffices to show that the lift of each cardinal from the smaller supremum
if bounded by the lift of some cardinal from the larger supremum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_supr_le_lift_supr'
{ι : Type v} {ι' : Type v'} {f : ι → cardinal.{v}} {f' : ι' → cardinal.{v'}}
(hf : bdd_above (range f)) (hf' : bdd_above (range f'))
(g : ι → ι') (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) :
lift.{v'} (supr f) ≤ lift.{v} (supr f') | lift_supr_le_lift_supr hf hf' h | lemma | cardinal.lift_supr_le_lift_supr' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"bdd_above",
"supr"
] | A variant of `lift_supr_le_lift_supr` with universes specialized via `w = v` and `w' = v'`.
This is sometimes necessary to avoid universe unification issues. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph_0 : cardinal.{u} | lift (#ℕ) | def | cardinal.aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | `ℵ₀` is the smallest infinite cardinal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_nat : #ℕ = ℵ₀ | (lift_id _).symm | lemma | cardinal.mk_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_ne_zero : ℵ₀ ≠ 0 | mk_ne_zero _ | theorem | cardinal.aleph_0_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_pos : 0 < ℵ₀ | pos_iff_ne_zero.2 aleph_0_ne_zero | theorem | cardinal.aleph_0_pos | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aleph_0 : lift ℵ₀ = ℵ₀ | lift_lift _ | theorem | cardinal.lift_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_lift {c : cardinal.{u}} : ℵ₀ ≤ lift.{v} c ↔ ℵ₀ ≤ c | by rw [←lift_aleph_0, lift_le] | theorem | cardinal.aleph_0_le_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_le_aleph_0 {c : cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ₀ | by rw [←lift_aleph_0, lift_le] | theorem | cardinal.lift_le_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_lt_lift {c : cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c | by rw [←lift_aleph_0, lift_lt] | theorem | cardinal.aleph_0_lt_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lt_aleph_0 {c : cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ | by rw [←lift_aleph_0, lift_lt] | theorem | cardinal.lift_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_fin (n : ℕ) : #(fin n) = n | by simp | theorem | cardinal.mk_fin | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_nat_cast (n : ℕ) : lift.{u} (n : cardinal.{v}) = n | by induction n; simp * | theorem | cardinal.lift_nat_cast | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_eq_nat_iff {a : cardinal.{u}} {n : ℕ} : lift.{v} a = n ↔ a = n | lift_injective.eq_iff' (lift_nat_cast n) | lemma | cardinal.lift_eq_nat_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_eq_lift_iff {n : ℕ} {a : cardinal.{u}} :
(n : cardinal) = lift.{v} a ↔ (n : cardinal) = a | by rw [←lift_nat_cast.{v} n, lift_inj] | lemma | cardinal.nat_eq_lift_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_le_nat_iff {a : cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n | by simp only [←lift_nat_cast, lift_le] | lemma | cardinal.lift_le_nat_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_le_lift_iff {n : ℕ} {a : cardinal.{u}} :
(n : cardinal) ≤ lift.{v} a ↔ (n : cardinal) ≤ a | by simp only [←lift_nat_cast, lift_le] | lemma | cardinal.nat_le_lift_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lt_nat_iff {a : cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n | by simp only [←lift_nat_cast, lift_lt] | lemma | cardinal.lift_lt_nat_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_lt_lift_iff {n : ℕ} {a : cardinal.{u}} :
(n : cardinal) < lift.{v} a ↔ (n : cardinal) < a | by simp only [←lift_nat_cast, lift_lt] | lemma | cardinal.nat_lt_lift_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mk_fin (n : ℕ) : lift (#(fin n)) = n | by simp | theorem | cardinal.lift_mk_fin | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe_finset {α : Type u} {s : finset α} : #s = ↑(finset.card s) | by simp | lemma | cardinal.mk_coe_finset | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finset",
"finset.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finset_of_fintype [fintype α] : #(finset α) = 2 ^ℕ fintype.card α | by simp | lemma | cardinal.mk_finset_of_fintype | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finset",
"fintype",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finsupp_lift_of_fintype (α : Type u) (β : Type v) [fintype α] [has_zero β] :
#(α →₀ β) = lift.{u} (#β) ^ℕ fintype.card α | by simpa using (@finsupp.equiv_fun_on_finite α β _ _).cardinal_eq | lemma | cardinal.mk_finsupp_lift_of_fintype | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finsupp.equiv_fun_on_finite",
"fintype",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finsupp_of_fintype (α β : Type u) [fintype α] [has_zero β] :
#(α →₀ β) = (#β) ^ℕ fintype.card α | by simp | lemma | cardinal.mk_finsupp_of_fintype | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"fintype",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_of_finset {α} (s : finset α) : (s.card : cardinal) ≤ #α | @mk_coe_finset _ s ▸ mk_set_le _ | theorem | cardinal.card_le_of_finset | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_pow {m n : ℕ} : (↑(pow m n) : cardinal) = m ^ n | by induction n; simp [pow_succ', power_add, *] | theorem | cardinal.nat_cast_pow | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_le {m n : ℕ} : (m : cardinal) ≤ n ↔ m ≤ n | by rw [← lift_mk_fin, ← lift_mk_fin, lift_le, le_def, function.embedding.nonempty_iff_card_le,
fintype.card_fin, fintype.card_fin] | theorem | cardinal.nat_cast_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"fintype.card_fin",
"function.embedding.nonempty_iff_card_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_lt {m n : ℕ} : (m : cardinal) < n ↔ m < n | by simp [lt_iff_le_not_le, ←not_le] | theorem | cardinal.nat_cast_lt | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_inj {m n : ℕ} : (m : cardinal) = n ↔ m = n | nat.cast_inj | theorem | cardinal.nat_cast_inj | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"nat.cast_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_injective : injective (coe : ℕ → cardinal) | nat.cast_injective | lemma | cardinal.nat_cast_injective | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"nat.cast_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_succ (n : ℕ) : (n.succ : cardinal) = succ n | (add_one_le_succ _).antisymm (succ_le_of_lt $ nat_cast_lt.2 $ nat.lt_succ_self _) | theorem | cardinal.nat_succ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_zero : succ (0 : cardinal) = 1 | by norm_cast | theorem | cardinal.succ_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_of {α : Type u} {n : ℕ} (H : ∀ s : finset α, s.card ≤ n) : # α ≤ n | begin
refine le_of_lt_succ (lt_of_not_ge $ λ hn, _),
rw [←cardinal.nat_succ, ←lift_mk_fin n.succ] at hn,
cases hn with f,
refine (H $ finset.univ.map f).not_lt _,
rw [finset.card_map, ←fintype.card, fintype.card_ulift, fintype.card_fin],
exact n.lt_succ_self
end | theorem | cardinal.card_le_of | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finset",
"finset.card_map",
"fintype.card_fin",
"fintype.card_ulift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cantor' (a) {b : cardinal} (hb : 1 < b) : a < b ^ a | begin
rw [←succ_le_iff, (by norm_cast : succ (1 : cardinal) = 2)] at hb,
exact (cantor a).trans_le (power_le_power_right hb)
end | theorem | cardinal.cantor' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_iff_pos {c : cardinal} : 1 ≤ c ↔ 0 < c | by rw [←succ_zero, succ_le_iff] | theorem | cardinal.one_le_iff_pos | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_iff_ne_zero {c : cardinal} : 1 ≤ c ↔ c ≠ 0 | by rw [one_le_iff_pos, pos_iff_ne_zero] | theorem | cardinal.one_le_iff_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_lt_aleph_0 (n : ℕ) : (n : cardinal.{u}) < ℵ₀ | succ_le_iff.1 begin
rw [←nat_succ, ←lift_mk_fin, aleph_0, lift_mk_le.{0 0 u}],
exact ⟨⟨coe, λ a b, fin.ext⟩⟩
end | theorem | cardinal.nat_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_aleph_0 : 1 < ℵ₀ | by simpa using nat_lt_aleph_0 1 | theorem | cardinal.one_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_aleph_0 : 1 ≤ ℵ₀ | one_lt_aleph_0.le | theorem | cardinal.one_le_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_aleph_0 {c : cardinal} : c < ℵ₀ ↔ ∃ n : ℕ, c = n | ⟨λ h, begin
rcases lt_lift_iff.1 h with ⟨c, rfl, h'⟩,
rcases le_mk_iff_exists_set.1 h'.1 with ⟨S, rfl⟩,
suffices : S.finite,
{ lift S to finset ℕ using this,
simp },
contrapose! h',
haveI := infinite.to_subtype h',
exact ⟨infinite.nat_embedding S⟩
end, λ ⟨n, e⟩, e.symm ▸ nat_lt_aleph_0 _⟩ | theorem | cardinal.lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"finset",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le {c : cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c | ⟨λ h n, (nat_lt_aleph_0 _).le.trans h,
λ h, le_of_not_lt $ λ hn, begin
rcases lt_aleph_0.1 hn with ⟨n, rfl⟩,
exact (nat.lt_succ_self _).not_le (nat_cast_le.1 (h (n+1)))
end⟩ | theorem | cardinal.aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_succ_limit_aleph_0 : is_succ_limit ℵ₀ | is_succ_limit_of_succ_lt $ λ a ha, begin
rcases lt_aleph_0.1 ha with ⟨n, rfl⟩,
rw ←nat_succ,
apply nat_lt_aleph_0
end | theorem | cardinal.is_succ_limit_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_aleph_0 : is_limit ℵ₀ | ⟨aleph_0_ne_zero, is_succ_limit_aleph_0⟩ | theorem | cardinal.is_limit_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.aleph_0_le {c : cardinal} (h : is_limit c) : ℵ₀ ≤ c | begin
by_contra' h',
rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩,
{ exact h.ne_zero.irrefl },
{ rw nat_succ at h,
exact not_is_succ_limit_succ _ h.is_succ_limit }
end | theorem | cardinal.is_limit.aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_nat_cast : range (coe : ℕ → cardinal) = Iio ℵ₀ | ext $ λ x, by simp only [mem_Iio, mem_range, eq_comm, lt_aleph_0] | lemma | cardinal.range_nat_cast | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_nat_iff {α : Type u} {n : ℕ} : #α = n ↔ nonempty (α ≃ fin n) | by rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq'] | theorem | cardinal.mk_eq_nat_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_aleph_0_iff_finite {α : Type u} : #α < ℵ₀ ↔ finite α | by simp only [lt_aleph_0, mk_eq_nat_iff, finite_iff_exists_equiv_fin] | theorem | cardinal.lt_aleph_0_iff_finite | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finite",
"finite_iff_exists_equiv_fin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_aleph_0_iff_fintype {α : Type u} : #α < ℵ₀ ↔ nonempty (fintype α) | lt_aleph_0_iff_finite.trans (finite_iff_nonempty_fintype _) | theorem | cardinal.lt_aleph_0_iff_fintype | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finite_iff_nonempty_fintype",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_aleph_0_of_finite (α : Type u) [finite α] : #α < ℵ₀ | lt_aleph_0_iff_finite.2 ‹_› | theorem | cardinal.lt_aleph_0_of_finite | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_aleph_0_iff_set_finite {S : set α} : #S < ℵ₀ ↔ S.finite | lt_aleph_0_iff_finite.trans finite_coe_iff | theorem | cardinal.lt_aleph_0_iff_set_finite | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_aleph_0_iff_subtype_finite {p : α → Prop} :
#{x // p x} < ℵ₀ ↔ {x | p x}.finite | lt_aleph_0_iff_set_finite | theorem | cardinal.lt_aleph_0_iff_subtype_finite | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_aleph_0_iff : #α ≤ ℵ₀ ↔ countable α | by rw [countable_iff_nonempty_embedding, aleph_0, ← lift_uzero (#α), lift_mk_le'] | lemma | cardinal.mk_le_aleph_0_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"countable",
"countable_iff_nonempty_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_aleph_0 [countable α] : #α ≤ ℵ₀ | mk_le_aleph_0_iff.mpr ‹_› | lemma | cardinal.mk_le_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_aleph_0_iff_set_countable {s : set α} : #s ≤ ℵ₀ ↔ s.countable | by rw [mk_le_aleph_0_iff, countable_coe_iff] | lemma | cardinal.le_aleph_0_iff_set_countable | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_aleph_0_iff_subtype_countable {p : α → Prop} :
#{x // p x} ≤ ℵ₀ ↔ {x | p x}.countable | le_aleph_0_iff_set_countable | lemma | cardinal.le_aleph_0_iff_subtype_countable | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
can_lift_cardinal_nat : can_lift cardinal ℕ coe (λ x, x < ℵ₀) | ⟨λ x hx, let ⟨n, hn⟩ := lt_aleph_0.mp hx in ⟨n, hn.symm⟩⟩ | instance | cardinal.can_lift_cardinal_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"can_lift",
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_aleph_0 {a b : cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a + b < ℵ₀ | match a, b, lt_aleph_0.1 ha, lt_aleph_0.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by rw [← nat.cast_add]; apply nat_lt_aleph_0
end | theorem | cardinal.add_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"nat.cast_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_aleph_0_iff {a b : cardinal} : a + b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ | ⟨λ h, ⟨(self_le_add_right _ _).trans_lt h, (self_le_add_left _ _).trans_lt h⟩,
λ ⟨h1, h2⟩, add_lt_aleph_0 h1 h2⟩ | lemma | cardinal.add_lt_aleph_0_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_add_iff {a b : cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b | by simp only [←not_lt, add_lt_aleph_0_iff, not_and_distrib] | lemma | cardinal.aleph_0_le_add_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"not_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_lt_aleph_0_iff {n : ℕ} {a : cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ | begin
cases n,
{ simpa using nat_lt_aleph_0 0 },
simp only [nat.succ_ne_zero, false_or],
induction n with n ih,
{ simp },
rw [succ_nsmul, add_lt_aleph_0_iff, ih, and_self]
end | lemma | cardinal.nsmul_lt_aleph_0_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"ih"
] | See also `cardinal.nsmul_lt_aleph_0_iff_of_ne_zero` if you already have `n ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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