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Theorem igamval 26380
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 2825 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ)))
2 fveq2 6839 . . . 4 (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴))
32oveq2d 7369 . . 3 (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴)))
41, 3ifbieq2d 4510 . 2 (𝑥 = 𝐴 → if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))
5 df-igam 26354 . 2 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
6 c0ex 11145 . . 3 0 ∈ V
7 ovex 7386 . . 3 (1 / (Γ‘𝐴)) ∈ V
86, 7ifex 4534 . 2 if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V
94, 5, 8fvmpt 6945 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 1541  wcel 2106  cdif 3905  ifcif 4484  cfv 6493  (class class class)co 7353  cc 11045  0cc0 11047  1c1 11048   / cdiv 11808  cn 12149  cz 12495  Γcgam 26350  1/Γcigam 26351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-mulcl 11109  ax-i2m1 11115
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-igam 26354
This theorem is referenced by:  igamz  26381  igamgam  26382  igamcl  26385
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