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Metamath Proof Explorer

Theorem mhm0

Description: A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015)

Ref Expression
Hypotheses mhm0.z ⊢ 0 = ( 0g ‘ 𝑆 )
mhm0.y ⊢ 𝑌 = ( 0g ‘ 𝑇 )
Assertion mhm0 ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → ( 𝐹 ‘ 0 ) = 𝑌 )

Proof

Step Hyp Ref Expression
1 mhm0.z ⊢ 0 = ( 0g ‘ 𝑆 )
2 mhm0.y ⊢ 𝑌 = ( 0g ‘ 𝑇 )
3 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
4 eqid ⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
5 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
6 eqid ⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 )
7 3 4 5 6 1 2 ismhm ⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ↔ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ) ∧ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ) )
8 7 simprbi ⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) )
9 8 simp3d ⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → ( 𝐹 ‘ 0 ) = 𝑌 )