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Metamath Proof Explorer

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 )