Step |
Hyp |
Ref |
Expression |
1 |
|
subg0.h |
⊢ 𝐻 = ( 𝐺 ↾s 𝑆 ) |
2 |
|
subginv.i |
⊢ 𝐼 = ( invg ‘ 𝐺 ) |
3 |
|
subginv.j |
⊢ 𝐽 = ( invg ‘ 𝐻 ) |
4 |
1
|
subggrp |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐻 ∈ Grp ) |
5 |
1
|
subgbas |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
6 |
5
|
eleq2d |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑋 ∈ 𝑆 ↔ 𝑋 ∈ ( Base ‘ 𝐻 ) ) ) |
7 |
6
|
biimpa |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∈ ( Base ‘ 𝐻 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
9 |
|
eqid |
⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 ) |
10 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
11 |
8 9 10 3
|
grprinv |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝐻 ) ) → ( 𝑋 ( +g ‘ 𝐻 ) ( 𝐽 ‘ 𝑋 ) ) = ( 0g ‘ 𝐻 ) ) |
12 |
4 7 11
|
syl2an2r |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ( +g ‘ 𝐻 ) ( 𝐽 ‘ 𝑋 ) ) = ( 0g ‘ 𝐻 ) ) |
13 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
14 |
1 13
|
ressplusg |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( +g ‘ 𝐺 ) = ( +g ‘ 𝐻 ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( +g ‘ 𝐺 ) = ( +g ‘ 𝐻 ) ) |
16 |
15
|
oveqd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐽 ‘ 𝑋 ) ) = ( 𝑋 ( +g ‘ 𝐻 ) ( 𝐽 ‘ 𝑋 ) ) ) |
17 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
18 |
1 17
|
subg0 |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
19 |
18
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
20 |
12 16 19
|
3eqtr4d |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐽 ‘ 𝑋 ) ) = ( 0g ‘ 𝐺 ) ) |
21 |
|
subgrcl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
22 |
21
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → 𝐺 ∈ Grp ) |
23 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
24 |
23
|
subgss |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
25 |
24
|
sselda |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∈ ( Base ‘ 𝐺 ) ) |
26 |
8 3
|
grpinvcl |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝐻 ) ) → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐻 ) ) |
27 |
26
|
ex |
⊢ ( 𝐻 ∈ Grp → ( 𝑋 ∈ ( Base ‘ 𝐻 ) → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐻 ) ) ) |
28 |
4 27
|
syl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑋 ∈ ( Base ‘ 𝐻 ) → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐻 ) ) ) |
29 |
5
|
eleq2d |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝐽 ‘ 𝑋 ) ∈ 𝑆 ↔ ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐻 ) ) ) |
30 |
28 6 29
|
3imtr4d |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑋 ∈ 𝑆 → ( 𝐽 ‘ 𝑋 ) ∈ 𝑆 ) ) |
31 |
30
|
imp |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( 𝐽 ‘ 𝑋 ) ∈ 𝑆 ) |
32 |
24
|
sselda |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐽 ‘ 𝑋 ) ∈ 𝑆 ) → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐺 ) ) |
33 |
31 32
|
syldan |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐺 ) ) |
34 |
23 13 17 2
|
grpinvid1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝐽 ‘ 𝑋 ) ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐽 ‘ 𝑋 ) ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐽 ‘ 𝑋 ) ) = ( 0g ‘ 𝐺 ) ) ) |
35 |
22 25 33 34
|
syl3anc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐽 ‘ 𝑋 ) ↔ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝐽 ‘ 𝑋 ) ) = ( 0g ‘ 𝐺 ) ) ) |
36 |
20 35
|
mpbird |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ) → ( 𝐼 ‘ 𝑋 ) = ( 𝐽 ‘ 𝑋 ) ) |