# Metamath Proof Explorer

## 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 = ( -g𝑆 )
ghmsub.n 𝑁 = ( -g𝑇 )
Assertion ghmsub ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈𝐵𝑉𝐵 ) → ( 𝐹 ‘ ( 𝑈 𝑉 ) ) = ( ( 𝐹𝑈 ) 𝑁 ( 𝐹𝑉 ) ) )

### Proof

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