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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 2827 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ))) | |
2 | fveq2 6907 | . . . 4 ⊢ (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴)) | |
3 | 2 | oveq2d 7447 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴))) |
4 | 1, 3 | ifbieq2d 4557 | . 2 ⊢ (𝑥 = 𝐴 → if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴)))) |
5 | df-igam 27079 | . 2 ⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥)))) | |
6 | c0ex 11253 | . . 3 ⊢ 0 ∈ V | |
7 | ovex 7464 | . . 3 ⊢ (1 / (Γ‘𝐴)) ∈ V | |
8 | 6, 7 | ifex 4581 | . 2 ⊢ if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V |
9 | 4, 5, 8 | fvmpt 7016 | 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 1537 ∈ wcel 2106 ∖ cdif 3960 ifcif 4531 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 / cdiv 11918 ℕcn 12264 ℤcz 12611 Γcgam 27075 1/Γcigam 27076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-igam 27079 |
This theorem is referenced by: igamz 27106 igamgam 27107 igamcl 27110 |
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