Theorem ghmsub

Description: Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014)

	Ref	Expression
Hypotheses	ghmsub.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	ghmsub.m	⊢ − = ( -g ‘ 𝑆 )
	ghmsub.n	⊢ 𝑁 = ( -g ‘ 𝑇 )
Assertion	ghmsub	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑈 − 𝑉 ) ) = ( ( 𝐹 ‘ 𝑈 ) 𝑁 ( 𝐹 ‘ 𝑉 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ghmsub.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
2		ghmsub.m	⊢ − = ( -g ‘ 𝑆 )
3		ghmsub.n	⊢ 𝑁 = ( -g ‘ 𝑇 )
4		ghmgrp1	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑆 ∈ Grp )
5	4	3ad2ant1	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → 𝑆 ∈ Grp )
6		simp3	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → 𝑉 ∈ 𝐵 )
7		eqid	⊢ ( invg ‘ 𝑆 ) = ( invg ‘ 𝑆 )
8	1 7	grpinvcl	⊢ ( ( 𝑆 ∈ Grp ∧ 𝑉 ∈ 𝐵 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ∈ 𝐵 )
9	5 6 8	syl2anc	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ∈ 𝐵 )
10		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
11		eqid	⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 )
12	1 10 11	ghmlin	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑈 ( +g ‘ 𝑆 ) ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) = ( ( 𝐹 ‘ 𝑈 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) )
13	9 12	syld3an3	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑈 ( +g ‘ 𝑆 ) ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) = ( ( 𝐹 ‘ 𝑈 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) )
14		eqid	⊢ ( invg ‘ 𝑇 ) = ( invg ‘ 𝑇 )
15	1 7 14	ghminv	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) = ( ( invg ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑉 ) ) )
16	15	3adant2	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) = ( ( invg ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑉 ) ) )
17	16	oveq2d	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑈 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) = ( ( 𝐹 ‘ 𝑈 ) ( +g ‘ 𝑇 ) ( ( invg ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑉 ) ) ) )
18	13 17	eqtrd	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑈 ( +g ‘ 𝑆 ) ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) = ( ( 𝐹 ‘ 𝑈 ) ( +g ‘ 𝑇 ) ( ( invg ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑉 ) ) ) )
19	1 10 7 2	grpsubval	⊢ ( ( 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝑈 − 𝑉 ) = ( 𝑈 ( +g ‘ 𝑆 ) ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) )
20	19	fveq2d	⊢ ( ( 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑈 − 𝑉 ) ) = ( 𝐹 ‘ ( 𝑈 ( +g ‘ 𝑆 ) ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) )
21	20	3adant1	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑈 − 𝑉 ) ) = ( 𝐹 ‘ ( 𝑈 ( +g ‘ 𝑆 ) ( ( invg ‘ 𝑆 ) ‘ 𝑉 ) ) ) )
22		eqid	⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
23	1 22	ghmf	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑇 ) )
24		ffvelrn	⊢ ( ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑇 ) ∧ 𝑈 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑈 ) ∈ ( Base ‘ 𝑇 ) )
25		ffvelrn	⊢ ( ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑇 ) ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑉 ) ∈ ( Base ‘ 𝑇 ) )
26	24 25	anim12dan	⊢ ( ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑇 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑈 ) ∈ ( Base ‘ 𝑇 ) ∧ ( 𝐹 ‘ 𝑉 ) ∈ ( Base ‘ 𝑇 ) ) )
27	23 26	sylan	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑈 ) ∈ ( Base ‘ 𝑇 ) ∧ ( 𝐹 ‘ 𝑉 ) ∈ ( Base ‘ 𝑇 ) ) )
28	27	3impb	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑈 ) ∈ ( Base ‘ 𝑇 ) ∧ ( 𝐹 ‘ 𝑉 ) ∈ ( Base ‘ 𝑇 ) ) )
29	22 11 14 3	grpsubval	⊢ ( ( ( 𝐹 ‘ 𝑈 ) ∈ ( Base ‘ 𝑇 ) ∧ ( 𝐹 ‘ 𝑉 ) ∈ ( Base ‘ 𝑇 ) ) → ( ( 𝐹 ‘ 𝑈 ) 𝑁 ( 𝐹 ‘ 𝑉 ) ) = ( ( 𝐹 ‘ 𝑈 ) ( +g ‘ 𝑇 ) ( ( invg ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑉 ) ) ) )
30	28 29	syl	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑈 ) 𝑁 ( 𝐹 ‘ 𝑉 ) ) = ( ( 𝐹 ‘ 𝑈 ) ( +g ‘ 𝑇 ) ( ( invg ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑉 ) ) ) )
31	18 21 30	3eqtr4d	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑈 − 𝑉 ) ) = ( ( 𝐹 ‘ 𝑈 ) 𝑁 ( 𝐹 ‘ 𝑉 ) ) )