Theorem dchrf

Description: A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016)

	Ref	Expression
Hypotheses	dchrmhm.g	⊢ 𝐺 = ( DChr ‘ 𝑁 )
	dchrmhm.z	⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
	dchrmhm.b	⊢ 𝐷 = ( Base ‘ 𝐺 )
	dchrf.b	⊢ 𝐵 = ( Base ‘ 𝑍 )
	dchrf.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
Assertion	dchrf	⊢ ( 𝜑 → 𝑋 : 𝐵 ⟶ ℂ )

Proof

Step	Hyp	Ref	Expression
1		dchrmhm.g	⊢ 𝐺 = ( DChr ‘ 𝑁 )
2		dchrmhm.z	⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
3		dchrmhm.b	⊢ 𝐷 = ( Base ‘ 𝐺 )
4		dchrf.b	⊢ 𝐵 = ( Base ‘ 𝑍 )
5		dchrf.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
6		eqid	⊢ ( Unit ‘ 𝑍 ) = ( Unit ‘ 𝑍 )
7	1 3	dchrrcl	⊢ ( 𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ )
8	5 7	syl	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
9	1 2 4 6 8 3	dchrelbas3	⊢ ( 𝜑 → ( 𝑋 ∈ 𝐷 ↔ ( 𝑋 : 𝐵 ⟶ ℂ ∧ ( ∀ 𝑥 ∈ ( Unit ‘ 𝑍 ) ∀ 𝑦 ∈ ( Unit ‘ 𝑍 ) ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( 𝑋 ‘ 𝑥 ) · ( 𝑋 ‘ 𝑦 ) ) ∧ ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑋 ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ ( Unit ‘ 𝑍 ) ) ) ) ) )
10	5 9	mpbid	⊢ ( 𝜑 → ( 𝑋 : 𝐵 ⟶ ℂ ∧ ( ∀ 𝑥 ∈ ( Unit ‘ 𝑍 ) ∀ 𝑦 ∈ ( Unit ‘ 𝑍 ) ( 𝑋 ‘ ( 𝑥 ( .r ‘ 𝑍 ) 𝑦 ) ) = ( ( 𝑋 ‘ 𝑥 ) · ( 𝑋 ‘ 𝑦 ) ) ∧ ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑋 ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ ( Unit ‘ 𝑍 ) ) ) ) )
11	10	simpld	⊢ ( 𝜑 → 𝑋 : 𝐵 ⟶ ℂ )