Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4100 | . . . 4 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → 𝐴 ∈ ℤ) | |
2 | 1 | zcnd 12076 | . . 3 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → 𝐴 ∈ ℂ) |
3 | igamval 25551 | . . 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 4469 | . 2 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) = 0) | |
6 | 4, 5 | eqtrd 2853 | 1 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ifcif 4463 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 / cdiv 11285 ℕcn 11626 ℤcz 11969 Γcgam 25521 1/Γcigam 25522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-neg 10861 df-z 11970 df-igam 25525 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |