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Metamath Proof Explorer

Theorem isghmd

Description: Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015)

Ref Expression
Hypotheses isghmd.x ⊢ 𝑋 = ( Base ‘ 𝑆 )
isghmd.y ⊢ 𝑌 = ( Base ‘ 𝑇 )
isghmd.a ⊢ + = ( +g ‘ 𝑆 )
isghmd.b ⊢ ⨣ = ( +g ‘ 𝑇 )
isghmd.s ⊢ ( 𝜑 → 𝑆 ∈ Grp )
isghmd.t ⊢ ( 𝜑 → 𝑇 ∈ Grp )
isghmd.f ⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑌 )
isghmd.l ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
Assertion isghmd ( 𝜑 → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )

Proof

Step Hyp Ref Expression
1 isghmd.x ⊢ 𝑋 = ( Base ‘ 𝑆 )
2 isghmd.y ⊢ 𝑌 = ( Base ‘ 𝑇 )
3 isghmd.a ⊢ + = ( +g ‘ 𝑆 )
4 isghmd.b ⊢ ⨣ = ( +g ‘ 𝑇 )
5 isghmd.s ⊢ ( 𝜑 → 𝑆 ∈ Grp )
6 isghmd.t ⊢ ( 𝜑 → 𝑇 ∈ Grp )
7 isghmd.f ⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝑌 )
8 isghmd.l ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
9 8 ralrimivva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
10 7 9 jca ⊢ ( 𝜑 → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ) )
11 1 2 3 4 isghm ⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ↔ ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ) ) )
12 5 6 10 11 syl21anbrc ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )