Definition df-igam 27082

Description: Define the inverse Gamma function, which is defined everywhere, unlike the Gamma function itself. (Contributed by Mario Carneiro, 16-Jul-2017.)

Assertion

Ref	Expression
df-igam	⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))

Detailed syntax breakdown of Definition df-igam

Step	Hyp	Ref	Expression
1		cigam 27079	. 2 class 1/Γ
2		vx	. . 3 setvar 𝑥
3		cc 11182	. . 3 class ℂ
4	2	cv 1536	. . . . 5 class 𝑥
5		cz 12639	. . . . . 6 class ℤ
6		cn 12293	. . . . . 6 class ℕ
7	5, 6	cdif 3973	. . . . 5 class (ℤ ∖ ℕ)
8	4, 7	wcel 2108	. . . 4 wff 𝑥 ∈ (ℤ ∖ ℕ)
9		cc0 11184	. . . 4 class 0
10		c1 11185	. . . . 5 class 1
11		cgam 27078	. . . . . 6 class Γ
12	4, 11	cfv 6573	. . . . 5 class (Γ‘𝑥)
13		cdiv 11947	. . . . 5 class /
14	10, 12, 13	co 7448	. . . 4 class (1 / (Γ‘𝑥))
15	8, 9, 14	cif 4548	. . 3 class if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥)))
16	2, 3, 15	cmpt 5249	. 2 class (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
17	1, 16	wceq 1537	1 wff 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))

This definition is referenced by:  igamval  27108  igamf  27112