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Theorem igamval 27091
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 2828 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ)))
2 fveq2 6905 . . . 4 (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴))
32oveq2d 7448 . . 3 (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴)))
41, 3ifbieq2d 4551 . 2 (𝑥 = 𝐴 → if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))
5 df-igam 27065 . 2 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
6 c0ex 11256 . . 3 0 ∈ V
7 ovex 7465 . . 3 (1 / (Γ‘𝐴)) ∈ V
86, 7ifex 4575 . 2 if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V
94, 5, 8fvmpt 7015 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 1539  wcel 2107  cdif 3947  ifcif 4524  cfv 6560  (class class class)co 7432  cc 11154  0cc0 11156  1c1 11157   / cdiv 11921  cn 12267  cz 12615  Γcgam 27061  1/Γcigam 27062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-mulcl 11218  ax-i2m1 11224
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-igam 27065
This theorem is referenced by:  igamz  27092  igamgam  27093  igamcl  27096
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