Step |
Hyp |
Ref |
Expression |
1 |
|
elnmz.1 |
⊢ 𝑁 = { 𝑥 ∈ 𝑋 ∣ ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 + 𝑦 ) ∈ 𝑆 ↔ ( 𝑦 + 𝑥 ) ∈ 𝑆 ) } |
2 |
|
nmzsubg.2 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
3 |
|
nmzsubg.3 |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
2
|
subgss |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ 𝑋 ) |
5 |
4
|
sselda |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑧 ∈ 𝑋 ) |
6 |
|
simpll |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
7 |
|
subgrcl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
8 |
6 7
|
syl |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → 𝐺 ∈ Grp ) |
9 |
6 4
|
syl |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → 𝑆 ⊆ 𝑋 ) |
10 |
|
simplrl |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → 𝑧 ∈ 𝑆 ) |
11 |
9 10
|
sseldd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → 𝑧 ∈ 𝑋 ) |
12 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
13 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
14 |
2 3 12 13
|
grplinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + 𝑧 ) = ( 0g ‘ 𝐺 ) ) |
15 |
8 11 14
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + 𝑧 ) = ( 0g ‘ 𝐺 ) ) |
16 |
15
|
oveq1d |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + 𝑧 ) + 𝑤 ) = ( ( 0g ‘ 𝐺 ) + 𝑤 ) ) |
17 |
13
|
subginvcl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑧 ∈ 𝑆 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑆 ) |
18 |
6 10 17
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑆 ) |
19 |
9 18
|
sseldd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑋 ) |
20 |
|
simplrr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → 𝑤 ∈ 𝑋 ) |
21 |
2 3
|
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ) ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + 𝑧 ) + 𝑤 ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + ( 𝑧 + 𝑤 ) ) ) |
22 |
8 19 11 20 21
|
syl13anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + 𝑧 ) + 𝑤 ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + ( 𝑧 + 𝑤 ) ) ) |
23 |
2 3 12
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) + 𝑤 ) = 𝑤 ) |
24 |
8 20 23
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( 0g ‘ 𝐺 ) + 𝑤 ) = 𝑤 ) |
25 |
16 22 24
|
3eqtr3d |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + ( 𝑧 + 𝑤 ) ) = 𝑤 ) |
26 |
|
simpr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( 𝑧 + 𝑤 ) ∈ 𝑆 ) |
27 |
3
|
subgcl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑆 ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + ( 𝑧 + 𝑤 ) ) ∈ 𝑆 ) |
28 |
6 18 26 27
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) + ( 𝑧 + 𝑤 ) ) ∈ 𝑆 ) |
29 |
25 28
|
eqeltrrd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → 𝑤 ∈ 𝑆 ) |
30 |
3
|
subgcl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑤 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) → ( 𝑤 + 𝑧 ) ∈ 𝑆 ) |
31 |
6 29 10 30
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑧 + 𝑤 ) ∈ 𝑆 ) → ( 𝑤 + 𝑧 ) ∈ 𝑆 ) |
32 |
|
simpll |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
33 |
|
simplrl |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → 𝑧 ∈ 𝑆 ) |
34 |
32 7
|
syl |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → 𝐺 ∈ Grp ) |
35 |
|
simplrr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → 𝑤 ∈ 𝑋 ) |
36 |
32 33 5
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → 𝑧 ∈ 𝑋 ) |
37 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
38 |
2 3 37
|
grppncan |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) → ( ( 𝑤 + 𝑧 ) ( -g ‘ 𝐺 ) 𝑧 ) = 𝑤 ) |
39 |
34 35 36 38
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → ( ( 𝑤 + 𝑧 ) ( -g ‘ 𝐺 ) 𝑧 ) = 𝑤 ) |
40 |
|
simpr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → ( 𝑤 + 𝑧 ) ∈ 𝑆 ) |
41 |
37
|
subgsubcl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝑤 + 𝑧 ) ( -g ‘ 𝐺 ) 𝑧 ) ∈ 𝑆 ) |
42 |
32 40 33 41
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → ( ( 𝑤 + 𝑧 ) ( -g ‘ 𝐺 ) 𝑧 ) ∈ 𝑆 ) |
43 |
39 42
|
eqeltrrd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → 𝑤 ∈ 𝑆 ) |
44 |
3
|
subgcl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) → ( 𝑧 + 𝑤 ) ∈ 𝑆 ) |
45 |
32 33 43 44
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) ∧ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) → ( 𝑧 + 𝑤 ) ∈ 𝑆 ) |
46 |
31 45
|
impbida |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( ( 𝑧 + 𝑤 ) ∈ 𝑆 ↔ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) ) |
47 |
46
|
anassrs |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑧 ∈ 𝑆 ) ∧ 𝑤 ∈ 𝑋 ) → ( ( 𝑧 + 𝑤 ) ∈ 𝑆 ↔ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) ) |
48 |
47
|
ralrimiva |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑧 ∈ 𝑆 ) → ∀ 𝑤 ∈ 𝑋 ( ( 𝑧 + 𝑤 ) ∈ 𝑆 ↔ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) ) |
49 |
1
|
elnmz |
⊢ ( 𝑧 ∈ 𝑁 ↔ ( 𝑧 ∈ 𝑋 ∧ ∀ 𝑤 ∈ 𝑋 ( ( 𝑧 + 𝑤 ) ∈ 𝑆 ↔ ( 𝑤 + 𝑧 ) ∈ 𝑆 ) ) ) |
50 |
5 48 49
|
sylanbrc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑧 ∈ 𝑁 ) |
51 |
50
|
ex |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑧 ∈ 𝑆 → 𝑧 ∈ 𝑁 ) ) |
52 |
51
|
ssrdv |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ 𝑁 ) |