Step |
Hyp |
Ref |
Expression |
1 |
|
grpidssd.m |
⊢ ( 𝜑 → 𝑀 ∈ Grp ) |
2 |
|
grpidssd.s |
⊢ ( 𝜑 → 𝑆 ∈ Grp ) |
3 |
|
grpidssd.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
4 |
|
grpidssd.c |
⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ 𝑀 ) ) |
5 |
|
grpidssd.o |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) |
6 |
|
eqid |
⊢ ( invg ‘ 𝑆 ) = ( invg ‘ 𝑆 ) |
7 |
3 6
|
grpinvcl |
⊢ ( ( 𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ∈ 𝐵 ) |
8 |
2 7
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ∈ 𝐵 ) |
9 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
10 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) |
11 |
|
oveq1 |
⊢ ( 𝑥 = ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑦 ) ) |
12 |
|
oveq1 |
⊢ ( 𝑥 = ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑦 ) ) |
13 |
11 12
|
eqeq12d |
⊢ ( 𝑥 = ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) → ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ↔ ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑦 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑦 ) ) ) |
14 |
|
oveq2 |
⊢ ( 𝑦 = 𝑋 → ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑦 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) ) |
15 |
|
oveq2 |
⊢ ( 𝑦 = 𝑋 → ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑦 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) ) |
16 |
14 15
|
eqeq12d |
⊢ ( 𝑦 = 𝑋 → ( ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑦 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑦 ) ↔ ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) ) ) |
17 |
13 16
|
rspc2va |
⊢ ( ( ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) ) |
18 |
8 9 10 17
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) ) |
19 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
20 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
21 |
3 19 20 6
|
grplinv |
⊢ ( ( 𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) = ( 0g ‘ 𝑆 ) ) |
22 |
2 21
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑆 ) 𝑋 ) = ( 0g ‘ 𝑆 ) ) |
23 |
4
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ ( Base ‘ 𝑀 ) ) |
24 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
25 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
26 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
27 |
|
eqid |
⊢ ( invg ‘ 𝑀 ) = ( invg ‘ 𝑀 ) |
28 |
24 25 26 27
|
grplinv |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝑀 ) ) → ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( 0g ‘ 𝑀 ) ) |
29 |
1 23 28
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( 0g ‘ 𝑀 ) ) |
30 |
1 2 3 4 5
|
grpidssd |
⊢ ( 𝜑 → ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑆 ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑆 ) ) |
32 |
29 31
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 0g ‘ 𝑆 ) = ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) ) |
33 |
18 22 32
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) ) |
34 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑀 ∈ Grp ) |
35 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝐵 ⊆ ( Base ‘ 𝑀 ) ) |
36 |
35 8
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑀 ) ) |
37 |
24 27
|
grpinvcl |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝑀 ) ) → ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑀 ) ) |
38 |
1 23 37
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑀 ) ) |
39 |
24 25
|
grprcan |
⊢ ( ( 𝑀 ∈ Grp ∧ ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑀 ) ∧ ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑀 ) ∧ 𝑋 ∈ ( Base ‘ 𝑀 ) ) ) → ( ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) ↔ ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) = ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) ) |
40 |
34 36 38 23 39
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) = ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑋 ) ↔ ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) = ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) ) |
41 |
33 40
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) = ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) |
42 |
41
|
ex |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 → ( ( invg ‘ 𝑆 ) ‘ 𝑋 ) = ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) ) |