Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubpropd2.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐺 ) ) |
2 |
|
grpsubpropd2.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐻 ) ) |
3 |
|
grpsubpropd2.3 |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
4 |
|
grpsubpropd2.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) ) |
5 |
|
simp1 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → 𝜑 ) |
6 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → 𝑎 ∈ ( Base ‘ 𝐺 ) ) |
7 |
1
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → 𝐵 = ( Base ‘ 𝐺 ) ) |
8 |
6 7
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → 𝑎 ∈ 𝐵 ) |
9 |
3
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → 𝐺 ∈ Grp ) |
10 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → 𝑏 ∈ ( Base ‘ 𝐺 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
12 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
13 |
11 12
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ∈ ( Base ‘ 𝐺 ) ) |
14 |
9 10 13
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ∈ ( Base ‘ 𝐺 ) ) |
15 |
14 7
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ∈ 𝐵 ) |
16 |
4
|
oveqrspc2v |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ∈ 𝐵 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) = ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) ) |
17 |
5 8 15 16
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) = ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) ) |
18 |
1 2 4
|
grpinvpropd |
⊢ ( 𝜑 → ( invg ‘ 𝐺 ) = ( invg ‘ 𝐻 ) ) |
19 |
18
|
fveq1d |
⊢ ( 𝜑 → ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) = ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) |
20 |
19
|
oveq2d |
⊢ ( 𝜑 → ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) = ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) |
21 |
20
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) = ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) |
22 |
17 21
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) = ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) |
23 |
22
|
mpoeq3dva |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Base ‘ 𝐺 ) , 𝑏 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) ) = ( 𝑎 ∈ ( Base ‘ 𝐺 ) , 𝑏 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) ) |
24 |
1 2
|
eqtr3d |
⊢ ( 𝜑 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) ) |
25 |
|
mpoeq12 |
⊢ ( ( ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) ∧ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) ) → ( 𝑎 ∈ ( Base ‘ 𝐺 ) , 𝑏 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) = ( 𝑎 ∈ ( Base ‘ 𝐻 ) , 𝑏 ∈ ( Base ‘ 𝐻 ) ↦ ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) ) |
26 |
24 24 25
|
syl2anc |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Base ‘ 𝐺 ) , 𝑏 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) = ( 𝑎 ∈ ( Base ‘ 𝐻 ) , 𝑏 ∈ ( Base ‘ 𝐻 ) ↦ ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) ) |
27 |
23 26
|
eqtrd |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Base ‘ 𝐺 ) , 𝑏 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) ) = ( 𝑎 ∈ ( Base ‘ 𝐻 ) , 𝑏 ∈ ( Base ‘ 𝐻 ) ↦ ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) ) |
28 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
29 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
30 |
11 28 12 29
|
grpsubfval |
⊢ ( -g ‘ 𝐺 ) = ( 𝑎 ∈ ( Base ‘ 𝐺 ) , 𝑏 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑏 ) ) ) |
31 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
32 |
|
eqid |
⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 ) |
33 |
|
eqid |
⊢ ( invg ‘ 𝐻 ) = ( invg ‘ 𝐻 ) |
34 |
|
eqid |
⊢ ( -g ‘ 𝐻 ) = ( -g ‘ 𝐻 ) |
35 |
31 32 33 34
|
grpsubfval |
⊢ ( -g ‘ 𝐻 ) = ( 𝑎 ∈ ( Base ‘ 𝐻 ) , 𝑏 ∈ ( Base ‘ 𝐻 ) ↦ ( 𝑎 ( +g ‘ 𝐻 ) ( ( invg ‘ 𝐻 ) ‘ 𝑏 ) ) ) |
36 |
27 30 35
|
3eqtr4g |
⊢ ( 𝜑 → ( -g ‘ 𝐺 ) = ( -g ‘ 𝐻 ) ) |