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Theorem igamval 25326
Description: Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
Assertion
Ref Expression
igamval (𝐴 ∈ ℂ → (1/Γ‘𝐴) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))

Proof of Theorem igamval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2854 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ)))
2 fveq2 6499 . . . 4 (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴))
32oveq2d 6992 . . 3 (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴)))
41, 3ifbieq2d 4375 . 2 (𝑥 = 𝐴 → if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))
5 df-igam 25300 . 2 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
6 c0ex 10433 . . 3 0 ∈ V
7 ovex 7008 . . 3 (1 / (Γ‘𝐴)) ∈ V
86, 7ifex 4398 . 2 if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V
94, 5, 8fvmpt 6595 1 (𝐴 ∈ ℂ → (1/Γ‘𝐴) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  cdif 3827  ifcif 4350  cfv 6188  (class class class)co 6976  cc 10333  0cc0 10335  1c1 10336   / cdiv 11098  cn 11439  cz 11793  Γcgam 25296  1/Γcigam 25297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-mulcl 10397  ax-i2m1 10403
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-igam 25300
This theorem is referenced by:  igamz  25327  igamgam  25328  igamcl  25331
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