Step |
Hyp |
Ref |
Expression |
1 |
|
grplsmid.p |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
2 |
|
subgrcl |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
3 |
2
|
adantr |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → 𝐺 ∈ Grp ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
5 |
4
|
subgss |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐴 ⊆ ( Base ‘ 𝐺 ) ) |
6 |
5
|
sselda |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → 𝑋 ∈ ( Base ‘ 𝐺 ) ) |
7 |
6
|
snssd |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → { 𝑋 } ⊆ ( Base ‘ 𝐺 ) ) |
8 |
5
|
adantr |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → 𝐴 ⊆ ( Base ‘ 𝐺 ) ) |
9 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
10 |
4 9 1
|
lsmelvalx |
⊢ ( ( 𝐺 ∈ Grp ∧ { 𝑋 } ⊆ ( Base ‘ 𝐺 ) ∧ 𝐴 ⊆ ( Base ‘ 𝐺 ) ) → ( 𝑥 ∈ ( { 𝑋 } ⊕ 𝐴 ) ↔ ∃ 𝑜 ∈ { 𝑋 } ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑜 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
11 |
3 7 8 10
|
syl3anc |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → ( 𝑥 ∈ ( { 𝑋 } ⊕ 𝐴 ) ↔ ∃ 𝑜 ∈ { 𝑋 } ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑜 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
12 |
|
oveq1 |
⊢ ( 𝑜 = 𝑋 → ( 𝑜 ( +g ‘ 𝐺 ) 𝑎 ) = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) |
13 |
12
|
eqeq2d |
⊢ ( 𝑜 = 𝑋 → ( 𝑥 = ( 𝑜 ( +g ‘ 𝐺 ) 𝑎 ) ↔ 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
14 |
13
|
rexbidv |
⊢ ( 𝑜 = 𝑋 → ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑜 ( +g ‘ 𝐺 ) 𝑎 ) ↔ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
15 |
14
|
rexsng |
⊢ ( 𝑋 ∈ 𝐴 → ( ∃ 𝑜 ∈ { 𝑋 } ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑜 ( +g ‘ 𝐺 ) 𝑎 ) ↔ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → ( ∃ 𝑜 ∈ { 𝑋 } ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑜 ( +g ‘ 𝐺 ) 𝑎 ) ↔ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
17 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) → 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) |
18 |
9
|
subgcl |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴 ) → ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ∈ 𝐴 ) |
19 |
18
|
ad4ant123 |
⊢ ( ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) → ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ∈ 𝐴 ) |
20 |
17 19
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) → 𝑥 ∈ 𝐴 ) |
21 |
20
|
r19.29an |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) → 𝑥 ∈ 𝐴 ) |
22 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ) |
23 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
24 |
23
|
subginvcl |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ∈ 𝐴 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ∈ 𝐴 ) |
26 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
27 |
9
|
subgcl |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝐴 ) |
28 |
22 25 26 27
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝐴 ) |
29 |
|
oveq2 |
⊢ ( 𝑎 = ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) → ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
30 |
29
|
eqeq2d |
⊢ ( 𝑎 = ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) → ( 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ↔ 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
31 |
30
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑎 = ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ) → ( 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ↔ 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
32 |
3
|
adantr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐺 ∈ Grp ) |
33 |
6
|
adantr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑋 ∈ ( Base ‘ 𝐺 ) ) |
34 |
8
|
sselda |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ( Base ‘ 𝐺 ) ) |
35 |
4 9 23
|
grpasscan1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑋 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ) = 𝑥 ) |
36 |
32 33 34 35
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ) = 𝑥 ) |
37 |
36
|
eqcomd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
38 |
28 31 37
|
rspcedvd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ) |
39 |
21 38
|
impbida |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑋 ( +g ‘ 𝐺 ) 𝑎 ) ↔ 𝑥 ∈ 𝐴 ) ) |
40 |
11 16 39
|
3bitrd |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → ( 𝑥 ∈ ( { 𝑋 } ⊕ 𝐴 ) ↔ 𝑥 ∈ 𝐴 ) ) |
41 |
40
|
eqrdv |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝐴 ) → ( { 𝑋 } ⊕ 𝐴 ) = 𝐴 ) |