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Mirrors > Home > MPE Home > Th. List > igamval | Structured version Visualization version GIF version |
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 2825 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ))) | |
2 | fveq2 6839 | . . . 4 ⊢ (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴)) | |
3 | 2 | oveq2d 7369 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴))) |
4 | 1, 3 | ifbieq2d 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 | |
8 | 6, 7 | ifex 4534 | . 2 ⊢ if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V |
9 | 4, 5, 8 | fvmpt 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 ℂcc 11045 0cc0 11047 1c1 11048 / cdiv 11808 ℕcn 12149 ℤcz 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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