| Step |
Hyp |
Ref |
Expression |
| 1 |
|
subgdprd.1 |
⊢ 𝐻 = ( 𝐺 ↾s 𝐴 ) |
| 2 |
|
reldmdprd |
⊢ Rel dom DProd |
| 3 |
2
|
brrelex2i |
⊢ ( 𝐻 dom DProd 𝑆 → 𝑆 ∈ V ) |
| 4 |
3
|
a1i |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐻 dom DProd 𝑆 → 𝑆 ∈ V ) ) |
| 5 |
2
|
brrelex2i |
⊢ ( 𝐺 dom DProd 𝑆 → 𝑆 ∈ V ) |
| 6 |
5
|
adantr |
⊢ ( ( 𝐺 dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) → 𝑆 ∈ V ) |
| 7 |
6
|
a1i |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝐺 dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) → 𝑆 ∈ V ) ) |
| 8 |
|
ffvelrn |
⊢ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ 𝑥 ∈ dom 𝑆 ) → ( 𝑆 ‘ 𝑥 ) ∈ ( SubGrp ‘ 𝐻 ) ) |
| 9 |
8
|
ad2ant2lr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑆 ‘ 𝑥 ) ∈ ( SubGrp ‘ 𝐻 ) ) |
| 10 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
| 11 |
10
|
subgss |
⊢ ( ( 𝑆 ‘ 𝑥 ) ∈ ( SubGrp ‘ 𝐻 ) → ( 𝑆 ‘ 𝑥 ) ⊆ ( Base ‘ 𝐻 ) ) |
| 12 |
9 11
|
syl |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑆 ‘ 𝑥 ) ⊆ ( Base ‘ 𝐻 ) ) |
| 13 |
1
|
subgbas |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐴 = ( Base ‘ 𝐻 ) ) |
| 14 |
13
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → 𝐴 = ( Base ‘ 𝐻 ) ) |
| 15 |
12 14
|
sseqtrrd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑆 ‘ 𝑥 ) ⊆ 𝐴 ) |
| 16 |
15
|
biantrud |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ↔ ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( 𝑆 ‘ 𝑥 ) ⊆ 𝐴 ) ) ) |
| 17 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 18 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) |
| 19 |
|
eldifi |
⊢ ( 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) → 𝑦 ∈ dom 𝑆 ) |
| 20 |
19
|
ad2antll |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → 𝑦 ∈ dom 𝑆 ) |
| 21 |
18 20
|
ffvelrnd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑆 ‘ 𝑦 ) ∈ ( SubGrp ‘ 𝐻 ) ) |
| 22 |
10
|
subgss |
⊢ ( ( 𝑆 ‘ 𝑦 ) ∈ ( SubGrp ‘ 𝐻 ) → ( 𝑆 ‘ 𝑦 ) ⊆ ( Base ‘ 𝐻 ) ) |
| 23 |
21 22
|
syl |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑆 ‘ 𝑦 ) ⊆ ( Base ‘ 𝐻 ) ) |
| 24 |
23 14
|
sseqtrrd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( 𝑆 ‘ 𝑦 ) ⊆ 𝐴 ) |
| 25 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
| 26 |
|
eqid |
⊢ ( Cntz ‘ 𝐻 ) = ( Cntz ‘ 𝐻 ) |
| 27 |
1 25 26
|
resscntz |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑆 ‘ 𝑦 ) ⊆ 𝐴 ) → ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) = ( ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∩ 𝐴 ) ) |
| 28 |
17 24 27
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) = ( ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∩ 𝐴 ) ) |
| 29 |
28
|
sseq2d |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ 𝑥 ) ⊆ ( ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∩ 𝐴 ) ) ) |
| 30 |
|
ssin |
⊢ ( ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( 𝑆 ‘ 𝑥 ) ⊆ 𝐴 ) ↔ ( 𝑆 ‘ 𝑥 ) ⊆ ( ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∩ 𝐴 ) ) |
| 31 |
29 30
|
bitr4di |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ↔ ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( 𝑆 ‘ 𝑥 ) ⊆ 𝐴 ) ) ) |
| 32 |
16 31
|
bitr4d |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ ( 𝑥 ∈ dom 𝑆 ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) ) → ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ) ) |
| 33 |
32
|
anassrs |
⊢ ( ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) ∧ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ) → ( ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ) ) |
| 34 |
33
|
ralbidva |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ) ) |
| 35 |
|
subgrcl |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
| 36 |
35
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → 𝐺 ∈ Grp ) |
| 37 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
| 38 |
37
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) ) |
| 39 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
| 40 |
36 38 39
|
3syl |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
| 41 |
1
|
subggrp |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐻 ∈ Grp ) |
| 42 |
41
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → 𝐻 ∈ Grp ) |
| 43 |
10
|
subgacs |
⊢ ( 𝐻 ∈ Grp → ( SubGrp ‘ 𝐻 ) ∈ ( ACS ‘ ( Base ‘ 𝐻 ) ) ) |
| 44 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐻 ) ∈ ( ACS ‘ ( Base ‘ 𝐻 ) ) → ( SubGrp ‘ 𝐻 ) ∈ ( Moore ‘ ( Base ‘ 𝐻 ) ) ) |
| 45 |
42 43 44
|
3syl |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( SubGrp ‘ 𝐻 ) ∈ ( Moore ‘ ( Base ‘ 𝐻 ) ) ) |
| 46 |
|
eqid |
⊢ ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) = ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) |
| 47 |
|
imassrn |
⊢ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ran 𝑆 |
| 48 |
|
frn |
⊢ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) → ran 𝑆 ⊆ ( SubGrp ‘ 𝐻 ) ) |
| 49 |
48
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ran 𝑆 ⊆ ( SubGrp ‘ 𝐻 ) ) |
| 50 |
47 49
|
sstrid |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( SubGrp ‘ 𝐻 ) ) |
| 51 |
|
mresspw |
⊢ ( ( SubGrp ‘ 𝐻 ) ∈ ( Moore ‘ ( Base ‘ 𝐻 ) ) → ( SubGrp ‘ 𝐻 ) ⊆ 𝒫 ( Base ‘ 𝐻 ) ) |
| 52 |
45 51
|
syl |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( SubGrp ‘ 𝐻 ) ⊆ 𝒫 ( Base ‘ 𝐻 ) ) |
| 53 |
50 52
|
sstrd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ 𝒫 ( Base ‘ 𝐻 ) ) |
| 54 |
|
sspwuni |
⊢ ( ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ 𝒫 ( Base ‘ 𝐻 ) ↔ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐻 ) ) |
| 55 |
53 54
|
sylib |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐻 ) ) |
| 56 |
45 46 55
|
mrcssidd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) |
| 57 |
46
|
mrccl |
⊢ ( ( ( SubGrp ‘ 𝐻 ) ∈ ( Moore ‘ ( Base ‘ 𝐻 ) ) ∧ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐻 ) ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ) |
| 58 |
45 55 57
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ) |
| 59 |
1
|
subsubg |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ↔ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ 𝐴 ) ) ) |
| 60 |
59
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ↔ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ 𝐴 ) ) ) |
| 61 |
58 60
|
mpbid |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ 𝐴 ) ) |
| 62 |
61
|
simpld |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 63 |
|
eqid |
⊢ ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
| 64 |
63
|
mrcsscl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∧ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) |
| 65 |
40 56 62 64
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) |
| 66 |
13
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → 𝐴 = ( Base ‘ 𝐻 ) ) |
| 67 |
55 66
|
sseqtrrd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ 𝐴 ) |
| 68 |
37
|
subgss |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐴 ⊆ ( Base ‘ 𝐺 ) ) |
| 69 |
68
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → 𝐴 ⊆ ( Base ‘ 𝐺 ) ) |
| 70 |
67 69
|
sstrd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐺 ) ) |
| 71 |
40 63 70
|
mrcssidd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) |
| 72 |
63
|
mrccl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐺 ) ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 73 |
40 70 72
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 74 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 75 |
63
|
mrcsscl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ 𝐴 ∧ 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ 𝐴 ) |
| 76 |
40 67 74 75
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ 𝐴 ) |
| 77 |
1
|
subsubg |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ↔ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ 𝐴 ) ) ) |
| 78 |
77
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ↔ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ 𝐴 ) ) ) |
| 79 |
73 76 78
|
mpbir2and |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ) |
| 80 |
46
|
mrcsscl |
⊢ ( ( ( SubGrp ‘ 𝐻 ) ∈ ( Moore ‘ ( Base ‘ 𝐻 ) ) ∧ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∧ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) |
| 81 |
45 71 79 80
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) |
| 82 |
65 81
|
eqssd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) = ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) |
| 83 |
82
|
ineq2d |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) ) |
| 84 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
| 85 |
1 84
|
subg0 |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
| 86 |
85
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
| 87 |
86
|
sneqd |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → { ( 0g ‘ 𝐺 ) } = { ( 0g ‘ 𝐻 ) } ) |
| 88 |
83 87
|
eqeq12d |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ↔ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) |
| 89 |
34 88
|
anbi12d |
⊢ ( ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) ∧ 𝑥 ∈ dom 𝑆 ) → ( ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ↔ ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) |
| 90 |
89
|
ralbidva |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ) → ( ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ↔ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) |
| 91 |
90
|
pm5.32da |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ↔ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) ) |
| 92 |
1
|
subsubg |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ⊆ 𝐴 ) ) ) |
| 93 |
|
elin |
⊢ ( 𝑥 ∈ ( ( SubGrp ‘ 𝐺 ) ∩ 𝒫 𝐴 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ 𝒫 𝐴 ) ) |
| 94 |
|
velpw |
⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 ) |
| 95 |
94
|
anbi2i |
⊢ ( ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ 𝒫 𝐴 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ⊆ 𝐴 ) ) |
| 96 |
93 95
|
bitri |
⊢ ( 𝑥 ∈ ( ( SubGrp ‘ 𝐺 ) ∩ 𝒫 𝐴 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ⊆ 𝐴 ) ) |
| 97 |
92 96
|
bitr4di |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ↔ 𝑥 ∈ ( ( SubGrp ‘ 𝐺 ) ∩ 𝒫 𝐴 ) ) ) |
| 98 |
97
|
eqrdv |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( SubGrp ‘ 𝐻 ) = ( ( SubGrp ‘ 𝐺 ) ∩ 𝒫 𝐴 ) ) |
| 99 |
98
|
sseq2d |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ran 𝑆 ⊆ ( SubGrp ‘ 𝐻 ) ↔ ran 𝑆 ⊆ ( ( SubGrp ‘ 𝐺 ) ∩ 𝒫 𝐴 ) ) ) |
| 100 |
|
ssin |
⊢ ( ( ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ↔ ran 𝑆 ⊆ ( ( SubGrp ‘ 𝐺 ) ∩ 𝒫 𝐴 ) ) |
| 101 |
99 100
|
bitr4di |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ran 𝑆 ⊆ ( SubGrp ‘ 𝐻 ) ↔ ( ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) |
| 102 |
101
|
anbi2d |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ ( SubGrp ‘ 𝐻 ) ) ↔ ( 𝑆 Fn dom 𝑆 ∧ ( ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) ) |
| 103 |
|
df-f |
⊢ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ↔ ( 𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ ( SubGrp ‘ 𝐻 ) ) ) |
| 104 |
|
df-f |
⊢ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ) ) |
| 105 |
104
|
anbi1i |
⊢ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ↔ ( ( 𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) |
| 106 |
|
anass |
⊢ ( ( ( 𝑆 Fn dom 𝑆 ∧ ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ↔ ( 𝑆 Fn dom 𝑆 ∧ ( ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) |
| 107 |
105 106
|
bitri |
⊢ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ↔ ( 𝑆 Fn dom 𝑆 ∧ ( ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) |
| 108 |
102 103 107
|
3bitr4g |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ↔ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) |
| 109 |
108
|
anbi1d |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ↔ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 110 |
91 109
|
bitr3d |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ↔ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 111 |
110
|
adantr |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ↔ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 112 |
|
dmexg |
⊢ ( 𝑆 ∈ V → dom 𝑆 ∈ V ) |
| 113 |
112
|
adantl |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → dom 𝑆 ∈ V ) |
| 114 |
|
eqidd |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → dom 𝑆 = dom 𝑆 ) |
| 115 |
41
|
adantr |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → 𝐻 ∈ Grp ) |
| 116 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
| 117 |
26 116 46
|
dmdprd |
⊢ ( ( dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆 ) → ( 𝐻 dom DProd 𝑆 ↔ ( 𝐻 ∈ Grp ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) ) |
| 118 |
|
3anass |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ↔ ( 𝐻 ∈ Grp ∧ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) ) |
| 119 |
117 118
|
bitrdi |
⊢ ( ( dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆 ) → ( 𝐻 dom DProd 𝑆 ↔ ( 𝐻 ∈ Grp ∧ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) ) ) |
| 120 |
119
|
baibd |
⊢ ( ( ( dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆 ) ∧ 𝐻 ∈ Grp ) → ( 𝐻 dom DProd 𝑆 ↔ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) ) |
| 121 |
113 114 115 120
|
syl21anc |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → ( 𝐻 dom DProd 𝑆 ↔ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐻 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐻 ) } ) ) ) ) |
| 122 |
35
|
adantr |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → 𝐺 ∈ Grp ) |
| 123 |
25 84 63
|
dmdprd |
⊢ ( ( dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆 ) → ( 𝐺 dom DProd 𝑆 ↔ ( 𝐺 ∈ Grp ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 124 |
|
3anass |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ↔ ( 𝐺 ∈ Grp ∧ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 125 |
123 124
|
bitrdi |
⊢ ( ( dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆 ) → ( 𝐺 dom DProd 𝑆 ↔ ( 𝐺 ∈ Grp ∧ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) ) |
| 126 |
125
|
baibd |
⊢ ( ( ( dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆 ) ∧ 𝐺 ∈ Grp ) → ( 𝐺 dom DProd 𝑆 ↔ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 127 |
113 114 122 126
|
syl21anc |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → ( 𝐺 dom DProd 𝑆 ↔ ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 128 |
127
|
anbi1d |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → ( ( 𝐺 dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ↔ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) |
| 129 |
|
an32 |
⊢ ( ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ↔ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) |
| 130 |
128 129
|
bitrdi |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → ( ( 𝐺 dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ↔ ( ( 𝑆 : dom 𝑆 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ∧ ∀ 𝑥 ∈ dom 𝑆 ( ∀ 𝑦 ∈ ( dom 𝑆 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( dom 𝑆 ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) ) ) |
| 131 |
111 121 130
|
3bitr4d |
⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ V ) → ( 𝐻 dom DProd 𝑆 ↔ ( 𝐺 dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) |
| 132 |
131
|
ex |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑆 ∈ V → ( 𝐻 dom DProd 𝑆 ↔ ( 𝐺 dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) ) |
| 133 |
4 7 132
|
pm5.21ndd |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐻 dom DProd 𝑆 ↔ ( 𝐺 dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴 ) ) ) |