Step |
Hyp |
Ref |
Expression |
1 |
|
qustriv.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
1
|
subgid |
⊢ ( 𝐺 ∈ Grp → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ) |
3 |
|
eqid |
⊢ ( 𝐺 ~QG 𝐵 ) = ( 𝐺 ~QG 𝐵 ) |
4 |
1 3
|
eqger |
⊢ ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ~QG 𝐵 ) Er 𝐵 ) |
5 |
|
errel |
⊢ ( ( 𝐺 ~QG 𝐵 ) Er 𝐵 → Rel ( 𝐺 ~QG 𝐵 ) ) |
6 |
2 4 5
|
3syl |
⊢ ( 𝐺 ∈ Grp → Rel ( 𝐺 ~QG 𝐵 ) ) |
7 |
|
relxp |
⊢ Rel ( 𝐵 × 𝐵 ) |
8 |
7
|
a1i |
⊢ ( 𝐺 ∈ Grp → Rel ( 𝐵 × 𝐵 ) ) |
9 |
|
df-3an |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) |
10 |
|
simpl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐺 ∈ Grp ) |
11 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
12 |
1 11
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
13 |
12
|
adantrr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
14 |
|
simprr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
15 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
16 |
1 15
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
17 |
10 13 14 16
|
syl3anc |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
18 |
17
|
ex |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) |
19 |
18
|
pm4.71d |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) ) |
20 |
9 19
|
bitr4id |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ) |
21 |
|
ssid |
⊢ 𝐵 ⊆ 𝐵 |
22 |
1 11 15 3
|
eqgval |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ⊆ 𝐵 ) → ( 𝑥 ( 𝐺 ~QG 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) ) |
23 |
21 22
|
mpan2 |
⊢ ( 𝐺 ∈ Grp → ( 𝑥 ( 𝐺 ~QG 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) ) |
24 |
|
brxp |
⊢ ( 𝑥 ( 𝐵 × 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) |
25 |
24
|
a1i |
⊢ ( 𝐺 ∈ Grp → ( 𝑥 ( 𝐵 × 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ) |
26 |
20 23 25
|
3bitr4d |
⊢ ( 𝐺 ∈ Grp → ( 𝑥 ( 𝐺 ~QG 𝐵 ) 𝑦 ↔ 𝑥 ( 𝐵 × 𝐵 ) 𝑦 ) ) |
27 |
6 8 26
|
eqbrrdv |
⊢ ( 𝐺 ∈ Grp → ( 𝐺 ~QG 𝐵 ) = ( 𝐵 × 𝐵 ) ) |