Theorem ghmmhm

Description: A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015)

	Ref	Expression
Assertion	ghmmhm	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		ghmgrp1	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑆 ∈ Grp )
2	1	grpmndd	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑆 ∈ Mnd )
3		ghmgrp2	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑇 ∈ Grp )
4	3	grpmndd	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑇 ∈ Mnd )
5		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
6		eqid	⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
7	5 6	ghmf	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
8		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
9		eqid	⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 )
10	5 8 9	ghmlin	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) )
11	10	3expb	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) )
12	11	ralrimivva	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) )
13		eqid	⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 )
14		eqid	⊢ ( 0g ‘ 𝑇 ) = ( 0g ‘ 𝑇 )
15	13 14	ghmid	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) )
16	7 12 15	3jca	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) )
17	5 6 8 9 13 14	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 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) ) )
18	2 4 16 17	syl21anbrc	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) )