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Mirrors > Home > MPE Home > Th. List > igamz | Structured version Visualization version GIF version |
Description: Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.) |
Ref | Expression |
---|---|
igamz | ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 4027 | . . . 4 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → 𝐴 ∈ ℤ) | |
2 | 1 | zcnd 12248 | . . 3 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → 𝐴 ∈ ℂ) |
3 | igamval 25883 | . . 3 ⊢ (𝐴 ∈ ℂ → (1/Γ‘𝐴) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴)))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴)))) |
5 | iftrue 4431 | . 2 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) = 0) | |
6 | 4, 5 | eqtrd 2771 | 1 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∖ cdif 3850 ifcif 4425 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 / cdiv 11454 ℕcn 11795 ℤcz 12141 Γcgam 25853 1/Γcigam 25854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-mulcl 10756 ax-i2m1 10762 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-neg 11030 df-z 12142 df-igam 25857 |
This theorem is referenced by: (None) |
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