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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 2854 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (ℤ ∖ ℕ) ↔ 𝐴 ∈ (ℤ ∖ ℕ))) | |
2 | fveq2 6499 | . . . 4 ⊢ (𝑥 = 𝐴 → (Γ‘𝑥) = (Γ‘𝐴)) | |
3 | 2 | oveq2d 6992 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (Γ‘𝑥)) = (1 / (Γ‘𝐴))) |
4 | 1, 3 | ifbieq2d 4375 | . 2 ⊢ (𝑥 = 𝐴 → if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴)))) |
5 | df-igam 25300 | . 2 ⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥)))) | |
6 | c0ex 10433 | . . 3 ⊢ 0 ∈ V | |
7 | ovex 7008 | . . 3 ⊢ (1 / (Γ‘𝐴)) ∈ V | |
8 | 6, 7 | ifex 4398 | . 2 ⊢ if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) ∈ V |
9 | 4, 5, 8 | fvmpt 6595 | 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 1507 ∈ wcel 2050 ∖ cdif 3827 ifcif 4350 ‘cfv 6188 (class class class)co 6976 ℂcc 10333 0cc0 10335 1c1 10336 / cdiv 11098 ℕcn 11439 ℤcz 11793 Γcgam 25296 1/Γcigam 25297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pr 5186 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-mulcl 10397 ax-i2m1 10403 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-ov 6979 df-igam 25300 |
This theorem is referenced by: igamz 25327 igamgam 25328 igamcl 25331 |
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