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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 ⊢ − = ( -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 ffvelcdm ⊢ ( ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑇 ) ∧ 𝑈 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑈 ) ∈ ( Base ‘ 𝑇 ) )
25 ffvelcdm ⊢ ( ( 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 ) ) )