Theorem mgmhmlin

Description: A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020)

	Ref	Expression
Hypotheses	mgmhmlin.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	mgmhmlin.p	⊢ + = ( +g ‘ 𝑆 )
	mgmhmlin.q	⊢ ⨣ = ( +g ‘ 𝑇 )
Assertion	mgmhmlin	⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mgmhmlin.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
2		mgmhmlin.p	⊢ + = ( +g ‘ 𝑆 )
3		mgmhmlin.q	⊢ ⨣ = ( +g ‘ 𝑇 )
4		eqid	⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
5	1 4 2 3	ismgmhm	⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ↔ ( ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ∧ ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ) ) )
6		fvoveq1	⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) )
7		fveq2	⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) )
8	7	oveq1d	⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
9	6 8	eqeq12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ) )
10		oveq2	⊢ ( 𝑦 = 𝑌 → ( 𝑋 + 𝑦 ) = ( 𝑋 + 𝑌 ) )
11	10	fveq2d	⊢ ( 𝑦 = 𝑌 → ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) )
12		fveq2	⊢ ( 𝑦 = 𝑌 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑌 ) )
13	12	oveq2d	⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) )
14	11 13	eqeq12d	⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ ( 𝑋 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) ) )
15	9 14	rspc2v	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) ) )
16	15	com12	⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) → ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) ) )
17	16	ad2antll	⊢ ( ( ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ∧ ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) ) ) → ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) ) )
18	5 17	sylbi	⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) ) )
19	18	3impib	⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ⨣ ( 𝐹 ‘ 𝑌 ) ) )