Step |
Hyp |
Ref |
Expression |
1 |
|
conjghm.x |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
conjghm.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
conjghm.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
conjsubg.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑆 ↦ ( ( 𝐴 + 𝑥 ) − 𝐴 ) ) |
5 |
|
conjnmz.1 |
⊢ 𝑁 = { 𝑦 ∈ 𝑋 ∣ ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ∈ 𝑆 ↔ ( 𝑧 + 𝑦 ) ∈ 𝑆 ) } |
6 |
5
|
ssrab3 |
⊢ 𝑁 ⊆ 𝑋 |
7 |
|
simpr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑁 ) → 𝐴 ∈ 𝑁 ) |
8 |
6 7
|
sselid |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑁 ) → 𝐴 ∈ 𝑋 ) |
9 |
1 2 3 4 5
|
conjnmz |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑁 ) → 𝑆 = ran 𝐹 ) |
10 |
8 9
|
jca |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑁 ) → ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) |
11 |
|
simprl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) → 𝐴 ∈ 𝑋 ) |
12 |
|
simplrr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) ∧ 𝑤 ∈ 𝑋 ) → 𝑆 = ran 𝐹 ) |
13 |
12
|
eleq2d |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) ∧ 𝑤 ∈ 𝑋 ) → ( ( 𝐴 + 𝑤 ) ∈ 𝑆 ↔ ( 𝐴 + 𝑤 ) ∈ ran 𝐹 ) ) |
14 |
|
subgrcl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
15 |
14
|
ad3antrrr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → 𝐺 ∈ Grp ) |
16 |
|
simpllr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → 𝐴 ∈ 𝑋 ) |
17 |
1
|
subgss |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ 𝑋 ) |
18 |
17
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) → 𝑆 ⊆ 𝑋 ) |
19 |
18
|
sselda |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝑋 ) |
20 |
1 2 3
|
grpaddsubass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝑥 ) − 𝐴 ) = ( 𝐴 + ( 𝑥 − 𝐴 ) ) ) |
21 |
15 16 19 16 20
|
syl13anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝐴 + 𝑥 ) − 𝐴 ) = ( 𝐴 + ( 𝑥 − 𝐴 ) ) ) |
22 |
21
|
eqeq1d |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( ( 𝐴 + 𝑥 ) − 𝐴 ) = ( 𝐴 + 𝑤 ) ↔ ( 𝐴 + ( 𝑥 − 𝐴 ) ) = ( 𝐴 + 𝑤 ) ) ) |
23 |
1 3
|
grpsubcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 − 𝐴 ) ∈ 𝑋 ) |
24 |
15 19 16 23
|
syl3anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 − 𝐴 ) ∈ 𝑋 ) |
25 |
|
simplr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑤 ∈ 𝑋 ) |
26 |
1 2
|
grplcan |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝑥 − 𝐴 ) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝐴 + ( 𝑥 − 𝐴 ) ) = ( 𝐴 + 𝑤 ) ↔ ( 𝑥 − 𝐴 ) = 𝑤 ) ) |
27 |
15 24 25 16 26
|
syl13anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝐴 + ( 𝑥 − 𝐴 ) ) = ( 𝐴 + 𝑤 ) ↔ ( 𝑥 − 𝐴 ) = 𝑤 ) ) |
28 |
1 2 3
|
grpsubadd |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ) ) → ( ( 𝑥 − 𝐴 ) = 𝑤 ↔ ( 𝑤 + 𝐴 ) = 𝑥 ) ) |
29 |
15 19 16 25 28
|
syl13anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝑥 − 𝐴 ) = 𝑤 ↔ ( 𝑤 + 𝐴 ) = 𝑥 ) ) |
30 |
22 27 29
|
3bitrd |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( ( 𝐴 + 𝑥 ) − 𝐴 ) = ( 𝐴 + 𝑤 ) ↔ ( 𝑤 + 𝐴 ) = 𝑥 ) ) |
31 |
|
eqcom |
⊢ ( ( 𝐴 + 𝑤 ) = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ↔ ( ( 𝐴 + 𝑥 ) − 𝐴 ) = ( 𝐴 + 𝑤 ) ) |
32 |
|
eqcom |
⊢ ( 𝑥 = ( 𝑤 + 𝐴 ) ↔ ( 𝑤 + 𝐴 ) = 𝑥 ) |
33 |
30 31 32
|
3bitr4g |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝐴 + 𝑤 ) = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ↔ 𝑥 = ( 𝑤 + 𝐴 ) ) ) |
34 |
33
|
rexbidva |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑋 ) → ( ∃ 𝑥 ∈ 𝑆 ( 𝐴 + 𝑤 ) = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ↔ ∃ 𝑥 ∈ 𝑆 𝑥 = ( 𝑤 + 𝐴 ) ) ) |
35 |
34
|
adantlrr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) ∧ 𝑤 ∈ 𝑋 ) → ( ∃ 𝑥 ∈ 𝑆 ( 𝐴 + 𝑤 ) = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ↔ ∃ 𝑥 ∈ 𝑆 𝑥 = ( 𝑤 + 𝐴 ) ) ) |
36 |
|
ovex |
⊢ ( 𝐴 + 𝑤 ) ∈ V |
37 |
|
eqeq1 |
⊢ ( 𝑦 = ( 𝐴 + 𝑤 ) → ( 𝑦 = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ↔ ( 𝐴 + 𝑤 ) = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ) ) |
38 |
37
|
rexbidv |
⊢ ( 𝑦 = ( 𝐴 + 𝑤 ) → ( ∃ 𝑥 ∈ 𝑆 𝑦 = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ↔ ∃ 𝑥 ∈ 𝑆 ( 𝐴 + 𝑤 ) = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ) ) |
39 |
4
|
rnmpt |
⊢ ran 𝐹 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝑆 𝑦 = ( ( 𝐴 + 𝑥 ) − 𝐴 ) } |
40 |
36 38 39
|
elab2 |
⊢ ( ( 𝐴 + 𝑤 ) ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑆 ( 𝐴 + 𝑤 ) = ( ( 𝐴 + 𝑥 ) − 𝐴 ) ) |
41 |
|
risset |
⊢ ( ( 𝑤 + 𝐴 ) ∈ 𝑆 ↔ ∃ 𝑥 ∈ 𝑆 𝑥 = ( 𝑤 + 𝐴 ) ) |
42 |
35 40 41
|
3bitr4g |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) ∧ 𝑤 ∈ 𝑋 ) → ( ( 𝐴 + 𝑤 ) ∈ ran 𝐹 ↔ ( 𝑤 + 𝐴 ) ∈ 𝑆 ) ) |
43 |
13 42
|
bitrd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) ∧ 𝑤 ∈ 𝑋 ) → ( ( 𝐴 + 𝑤 ) ∈ 𝑆 ↔ ( 𝑤 + 𝐴 ) ∈ 𝑆 ) ) |
44 |
43
|
ralrimiva |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) → ∀ 𝑤 ∈ 𝑋 ( ( 𝐴 + 𝑤 ) ∈ 𝑆 ↔ ( 𝑤 + 𝐴 ) ∈ 𝑆 ) ) |
45 |
5
|
elnmz |
⊢ ( 𝐴 ∈ 𝑁 ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑤 ∈ 𝑋 ( ( 𝐴 + 𝑤 ) ∈ 𝑆 ↔ ( 𝑤 + 𝐴 ) ∈ 𝑆 ) ) ) |
46 |
11 44 45
|
sylanbrc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) → 𝐴 ∈ 𝑁 ) |
47 |
10 46
|
impbida |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐴 ∈ 𝑁 ↔ ( 𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹 ) ) ) |