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Theorem igamval 26999
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 2817 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ)))
2 fveq2 6902 . . . 4 (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴))
32oveq2d 7442 . . 3 (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴)))
41, 3ifbieq2d 4558 . 2 (𝑥 = 𝐴 → if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))
5 df-igam 26973 . 2 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
6 c0ex 11246 . . 3 0 ∈ V
7 ovex 7459 . . 3 (1 / (Γ‘𝐴)) ∈ V
86, 7ifex 4582 . 2 if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V
94, 5, 8fvmpt 7010 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 1533  wcel 2098  cdif 3946  ifcif 4532  cfv 6553  (class class class)co 7426  cc 11144  0cc0 11146  1c1 11147   / cdiv 11909  cn 12250  cz 12596  Γcgam 26969  1/Γcigam 26970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-mulcl 11208  ax-i2m1 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-igam 26973
This theorem is referenced by:  igamz  27000  igamgam  27001  igamcl  27004
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