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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 3955 | . . . 4 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → 𝐴 ∈ ℤ) | |
2 | 1 | zcnd 11840 | . . 3 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → 𝐴 ∈ ℂ) |
3 | igamval 25236 | . . 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 4313 | . 2 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))) = 0) | |
6 | 4, 5 | eqtrd 2814 | 1 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∖ cdif 3789 ifcif 4307 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 0cc0 10274 1c1 10275 / cdiv 11035 ℕcn 11379 ℤcz 11733 Γcgam 25206 1/Γcigam 25207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-mulcl 10336 ax-i2m1 10342 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-iota 6101 df-fun 6139 df-fv 6145 df-ov 6927 df-neg 10611 df-z 11734 df-igam 25210 |
This theorem is referenced by: (None) |
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