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| Description: Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.) | 
| Ref | Expression | 
|---|---|
| igamval | ⊢ (𝐴 ∈ ℂ → (1/Γ‘𝐴) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eleq1 2828 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ))) | |
| 2 | fveq2 6905 | . . . 4 ⊢ (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴)) | |
| 3 | 2 | oveq2d 7448 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴))) | 
| 4 | 1, 3 | ifbieq2d 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 | |
| 8 | 6, 7 | ifex 4575 | . 2 ⊢ if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V | 
| 9 | 4, 5, 8 | fvmpt 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 ℂcc 11154 0cc0 11156 1c1 11157 / cdiv 11921 ℕcn 12267 ℤcz 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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