Step |
Hyp |
Ref |
Expression |
1 |
|
dprdsplit.2 |
⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
2 |
|
dprdsplit.i |
⊢ ( 𝜑 → ( 𝐶 ∩ 𝐷 ) = ∅ ) |
3 |
|
dprdsplit.u |
⊢ ( 𝜑 → 𝐼 = ( 𝐶 ∪ 𝐷 ) ) |
4 |
|
dmdprdsplit.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
5 |
|
dmdprdsplit.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
6 |
|
dmdprdsplit2.1 |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) ) |
7 |
|
dmdprdsplit2.2 |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑆 ↾ 𝐷 ) ) |
8 |
|
dmdprdsplit2.3 |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
9 |
|
dmdprdsplit2.4 |
⊢ ( 𝜑 → ( ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) = { 0 } ) |
10 |
|
dmdprdsplit2lem.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
11 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 𝐼 = ( 𝐶 ∪ 𝐷 ) ) |
12 |
11
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑌 ∈ 𝐼 ↔ 𝑌 ∈ ( 𝐶 ∪ 𝐷 ) ) ) |
13 |
|
elun |
⊢ ( 𝑌 ∈ ( 𝐶 ∪ 𝐷 ) ↔ ( 𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷 ) ) |
14 |
12 13
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑌 ∈ 𝐼 ↔ ( 𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷 ) ) ) |
15 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) ) |
16 |
|
ssun1 |
⊢ 𝐶 ⊆ ( 𝐶 ∪ 𝐷 ) |
17 |
16 3
|
sseqtrrid |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐼 ) |
18 |
1 17
|
fssresd |
⊢ ( 𝜑 → ( 𝑆 ↾ 𝐶 ) : 𝐶 ⟶ ( SubGrp ‘ 𝐺 ) ) |
19 |
18
|
fdmd |
⊢ ( 𝜑 → dom ( 𝑆 ↾ 𝐶 ) = 𝐶 ) |
20 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → dom ( 𝑆 ↾ 𝐶 ) = 𝐶 ) |
21 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑋 ∈ 𝐶 ) |
22 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑌 ∈ 𝐶 ) |
23 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑋 ≠ 𝑌 ) |
24 |
15 20 21 22 23 4
|
dprdcntz |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑌 ) ) ) |
25 |
|
fvres |
⊢ ( 𝑋 ∈ 𝐶 → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) = ( 𝑆 ‘ 𝑋 ) ) |
26 |
25
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) = ( 𝑆 ‘ 𝑋 ) ) |
27 |
|
fvres |
⊢ ( 𝑌 ∈ 𝐶 → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑌 ) = ( 𝑆 ‘ 𝑌 ) ) |
28 |
27
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑌 ) = ( 𝑆 ‘ 𝑌 ) ) |
29 |
28
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑍 ‘ ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑌 ) ) = ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) |
30 |
24 26 29
|
3sstr3d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) |
31 |
30
|
exp32 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑌 ∈ 𝐶 → ( 𝑋 ≠ 𝑌 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) ) ) |
32 |
25
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) = ( 𝑆 ‘ 𝑋 ) ) |
33 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) ) |
34 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → dom ( 𝑆 ↾ 𝐶 ) = 𝐶 ) |
35 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑋 ∈ 𝐶 ) |
36 |
33 34 35
|
dprdub |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) |
37 |
32 36
|
eqsstrrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) |
38 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
39 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
40 |
39
|
dprdssv |
⊢ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ⊆ ( Base ‘ 𝐺 ) |
41 |
|
fvres |
⊢ ( 𝑌 ∈ 𝐷 → ( ( 𝑆 ↾ 𝐷 ) ‘ 𝑌 ) = ( 𝑆 ‘ 𝑌 ) ) |
42 |
41
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑆 ↾ 𝐷 ) ‘ 𝑌 ) = ( 𝑆 ‘ 𝑌 ) ) |
43 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝐺 dom DProd ( 𝑆 ↾ 𝐷 ) ) |
44 |
|
ssun2 |
⊢ 𝐷 ⊆ ( 𝐶 ∪ 𝐷 ) |
45 |
44 3
|
sseqtrrid |
⊢ ( 𝜑 → 𝐷 ⊆ 𝐼 ) |
46 |
1 45
|
fssresd |
⊢ ( 𝜑 → ( 𝑆 ↾ 𝐷 ) : 𝐷 ⟶ ( SubGrp ‘ 𝐺 ) ) |
47 |
46
|
fdmd |
⊢ ( 𝜑 → dom ( 𝑆 ↾ 𝐷 ) = 𝐷 ) |
48 |
47
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → dom ( 𝑆 ↾ 𝐷 ) = 𝐷 ) |
49 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑌 ∈ 𝐷 ) |
50 |
43 48 49
|
dprdub |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑆 ↾ 𝐷 ) ‘ 𝑌 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) |
51 |
42 50
|
eqsstrrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑆 ‘ 𝑌 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) |
52 |
39 4
|
cntz2ss |
⊢ ( ( ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ⊆ ( Base ‘ 𝐺 ) ∧ ( 𝑆 ‘ 𝑌 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) → ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) |
53 |
40 51 52
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) |
54 |
38 53
|
sstrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) |
55 |
37 54
|
sstrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) |
56 |
55
|
exp32 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑌 ∈ 𝐷 → ( 𝑋 ≠ 𝑌 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) ) ) |
57 |
31 56
|
jaod |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷 ) → ( 𝑋 ≠ 𝑌 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) ) ) |
58 |
14 57
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑌 ∈ 𝐼 → ( 𝑋 ≠ 𝑌 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) ) ) |
59 |
|
dprdgrp |
⊢ ( 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) → 𝐺 ∈ Grp ) |
60 |
6 59
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
61 |
60
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 𝐺 ∈ Grp ) |
62 |
39
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) ) |
63 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
64 |
61 62 63
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
65 |
|
difundir |
⊢ ( ( 𝐶 ∪ 𝐷 ) ∖ { 𝑋 } ) = ( ( 𝐶 ∖ { 𝑋 } ) ∪ ( 𝐷 ∖ { 𝑋 } ) ) |
66 |
11
|
difeq1d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐼 ∖ { 𝑋 } ) = ( ( 𝐶 ∪ 𝐷 ) ∖ { 𝑋 } ) ) |
67 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 𝑋 ∈ 𝐶 ) |
68 |
67
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → { 𝑋 } ⊆ 𝐶 ) |
69 |
|
sslin |
⊢ ( { 𝑋 } ⊆ 𝐶 → ( 𝐷 ∩ { 𝑋 } ) ⊆ ( 𝐷 ∩ 𝐶 ) ) |
70 |
68 69
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐷 ∩ { 𝑋 } ) ⊆ ( 𝐷 ∩ 𝐶 ) ) |
71 |
|
incom |
⊢ ( 𝐶 ∩ 𝐷 ) = ( 𝐷 ∩ 𝐶 ) |
72 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐶 ∩ 𝐷 ) = ∅ ) |
73 |
71 72
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐷 ∩ 𝐶 ) = ∅ ) |
74 |
|
sseq0 |
⊢ ( ( ( 𝐷 ∩ { 𝑋 } ) ⊆ ( 𝐷 ∩ 𝐶 ) ∧ ( 𝐷 ∩ 𝐶 ) = ∅ ) → ( 𝐷 ∩ { 𝑋 } ) = ∅ ) |
75 |
70 73 74
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐷 ∩ { 𝑋 } ) = ∅ ) |
76 |
|
disj3 |
⊢ ( ( 𝐷 ∩ { 𝑋 } ) = ∅ ↔ 𝐷 = ( 𝐷 ∖ { 𝑋 } ) ) |
77 |
75 76
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 𝐷 = ( 𝐷 ∖ { 𝑋 } ) ) |
78 |
77
|
uneq2d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝐶 ∖ { 𝑋 } ) ∪ 𝐷 ) = ( ( 𝐶 ∖ { 𝑋 } ) ∪ ( 𝐷 ∖ { 𝑋 } ) ) ) |
79 |
65 66 78
|
3eqtr4a |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐼 ∖ { 𝑋 } ) = ( ( 𝐶 ∖ { 𝑋 } ) ∪ 𝐷 ) ) |
80 |
79
|
imaeq2d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) = ( 𝑆 “ ( ( 𝐶 ∖ { 𝑋 } ) ∪ 𝐷 ) ) ) |
81 |
|
imaundi |
⊢ ( 𝑆 “ ( ( 𝐶 ∖ { 𝑋 } ) ∪ 𝐷 ) ) = ( ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ( 𝑆 “ 𝐷 ) ) |
82 |
80 81
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) = ( ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ( 𝑆 “ 𝐷 ) ) ) |
83 |
82
|
unieqd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) = ∪ ( ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ( 𝑆 “ 𝐷 ) ) ) |
84 |
|
uniun |
⊢ ∪ ( ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ( 𝑆 “ 𝐷 ) ) = ( ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ∪ ( 𝑆 “ 𝐷 ) ) |
85 |
83 84
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) = ( ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ∪ ( 𝑆 “ 𝐷 ) ) ) |
86 |
|
difss |
⊢ ( 𝐶 ∖ { 𝑋 } ) ⊆ 𝐶 |
87 |
|
imass2 |
⊢ ( ( 𝐶 ∖ { 𝑋 } ) ⊆ 𝐶 → ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( 𝑆 “ 𝐶 ) ) |
88 |
|
uniss |
⊢ ( ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( 𝑆 “ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ∪ ( 𝑆 “ 𝐶 ) ) |
89 |
86 87 88
|
mp2b |
⊢ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ∪ ( 𝑆 “ 𝐶 ) |
90 |
|
imassrn |
⊢ ( 𝑆 “ 𝐶 ) ⊆ ran 𝑆 |
91 |
1
|
frnd |
⊢ ( 𝜑 → ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ) |
92 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ) |
93 |
|
mresspw |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
94 |
64 93
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( SubGrp ‘ 𝐺 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
95 |
92 94
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ran 𝑆 ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
96 |
90 95
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 “ 𝐶 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
97 |
|
sspwuni |
⊢ ( ( 𝑆 “ 𝐶 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ↔ ∪ ( 𝑆 “ 𝐶 ) ⊆ ( Base ‘ 𝐺 ) ) |
98 |
96 97
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ 𝐶 ) ⊆ ( Base ‘ 𝐺 ) ) |
99 |
89 98
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( Base ‘ 𝐺 ) ) |
100 |
64 10 99
|
mrcssidd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) |
101 |
|
imassrn |
⊢ ( 𝑆 “ 𝐷 ) ⊆ ran 𝑆 |
102 |
101 95
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 “ 𝐷 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
103 |
|
sspwuni |
⊢ ( ( 𝑆 “ 𝐷 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ↔ ∪ ( 𝑆 “ 𝐷 ) ⊆ ( Base ‘ 𝐺 ) ) |
104 |
102 103
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ 𝐷 ) ⊆ ( Base ‘ 𝐺 ) ) |
105 |
64 10 104
|
mrcssidd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ 𝐷 ) ⊆ ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐷 ) ) ) |
106 |
10
|
dprdspan |
⊢ ( 𝐺 dom DProd ( 𝑆 ↾ 𝐷 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) = ( 𝐾 ‘ ∪ ran ( 𝑆 ↾ 𝐷 ) ) ) |
107 |
7 106
|
syl |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) = ( 𝐾 ‘ ∪ ran ( 𝑆 ↾ 𝐷 ) ) ) |
108 |
|
df-ima |
⊢ ( 𝑆 “ 𝐷 ) = ran ( 𝑆 ↾ 𝐷 ) |
109 |
108
|
unieqi |
⊢ ∪ ( 𝑆 “ 𝐷 ) = ∪ ran ( 𝑆 ↾ 𝐷 ) |
110 |
109
|
fveq2i |
⊢ ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐷 ) ) = ( 𝐾 ‘ ∪ ran ( 𝑆 ↾ 𝐷 ) ) |
111 |
107 110
|
eqtr4di |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) = ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐷 ) ) ) |
112 |
111
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) = ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐷 ) ) ) |
113 |
105 112
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ 𝐷 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) |
114 |
|
unss12 |
⊢ ( ( ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∧ ∪ ( 𝑆 “ 𝐷 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) → ( ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ∪ ( 𝑆 “ 𝐷 ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∪ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
115 |
100 113 114
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ∪ ( 𝑆 “ 𝐷 ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∪ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
116 |
10
|
mrccl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( Base ‘ 𝐺 ) ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
117 |
64 99 116
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
118 |
|
dprdsubg |
⊢ ( 𝐺 dom DProd ( 𝑆 ↾ 𝐷 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
119 |
7 118
|
syl |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
120 |
119
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
121 |
|
eqid |
⊢ ( LSSum ‘ 𝐺 ) = ( LSSum ‘ 𝐺 ) |
122 |
121
|
lsmunss |
⊢ ( ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∪ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
123 |
117 120 122
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∪ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
124 |
115 123
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ∪ ∪ ( 𝑆 “ 𝐷 ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
125 |
85 124
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
126 |
89
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ∪ ( 𝑆 “ 𝐶 ) ) |
127 |
64 10 126 98
|
mrcssd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐶 ) ) ) |
128 |
10
|
dprdspan |
⊢ ( 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) = ( 𝐾 ‘ ∪ ran ( 𝑆 ↾ 𝐶 ) ) ) |
129 |
6 128
|
syl |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) = ( 𝐾 ‘ ∪ ran ( 𝑆 ↾ 𝐶 ) ) ) |
130 |
|
df-ima |
⊢ ( 𝑆 “ 𝐶 ) = ran ( 𝑆 ↾ 𝐶 ) |
131 |
130
|
unieqi |
⊢ ∪ ( 𝑆 “ 𝐶 ) = ∪ ran ( 𝑆 ↾ 𝐶 ) |
132 |
131
|
fveq2i |
⊢ ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐶 ) ) = ( 𝐾 ‘ ∪ ran ( 𝑆 ↾ 𝐶 ) ) |
133 |
129 132
|
eqtr4di |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) = ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐶 ) ) ) |
134 |
133
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) = ( 𝐾 ‘ ∪ ( 𝑆 “ 𝐶 ) ) ) |
135 |
127 134
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) |
136 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
137 |
135 136
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
138 |
121 4
|
lsmsubg |
⊢ ( ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) → ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
139 |
117 120 137 138
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
140 |
10
|
mrcsscl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ∧ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
141 |
64 125 139 140
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
142 |
|
sslin |
⊢ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ⊆ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) → ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ⊆ ( ( 𝑆 ‘ 𝑋 ) ∩ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) ) |
143 |
141 142
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ⊆ ( ( 𝑆 ‘ 𝑋 ) ∩ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) ) |
144 |
17
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 𝑋 ∈ 𝐼 ) |
145 |
1
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐼 ) → ( 𝑆 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
146 |
144 145
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
147 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) = ( 𝑆 ‘ 𝑋 ) ) |
148 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) ) |
149 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → dom ( 𝑆 ↾ 𝐶 ) = 𝐶 ) |
150 |
148 149 67
|
dprdub |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) |
151 |
147 150
|
eqsstrrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) |
152 |
|
dprdsubg |
⊢ ( 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
153 |
6 152
|
syl |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
154 |
153
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
155 |
121
|
lsmlub |
⊢ ( ( ( 𝑆 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∧ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) ↔ ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) ) |
156 |
146 117 154 155
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∧ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) ↔ ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) ) |
157 |
151 135 156
|
mpbi2and |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ⊆ ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ) |
158 |
157
|
ssrind |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ⊆ ( ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
159 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝐺 DProd ( 𝑆 ↾ 𝐶 ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) = { 0 } ) |
160 |
158 159
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ⊆ { 0 } ) |
161 |
121
|
lsmub1 |
⊢ ( ( ( 𝑆 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ) |
162 |
146 117 161
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ) |
163 |
5
|
subg0cl |
⊢ ( ( 𝑆 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ ( 𝑆 ‘ 𝑋 ) ) |
164 |
146 163
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 0 ∈ ( 𝑆 ‘ 𝑋 ) ) |
165 |
162 164
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 0 ∈ ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ) |
166 |
5
|
subg0cl |
⊢ ( ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) |
167 |
120 166
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 0 ∈ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) |
168 |
165 167
|
elind |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 0 ∈ ( ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
169 |
168
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → { 0 } ⊆ ( ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) |
170 |
160 169
|
eqssd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( ( 𝑆 ‘ 𝑋 ) ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) = { 0 } ) |
171 |
|
resima2 |
⊢ ( ( 𝐶 ∖ { 𝑋 } ) ⊆ 𝐶 → ( ( 𝑆 ↾ 𝐶 ) “ ( 𝐶 ∖ { 𝑋 } ) ) = ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) |
172 |
86 171
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ↾ 𝐶 ) “ ( 𝐶 ∖ { 𝑋 } ) ) = ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) |
173 |
172
|
unieqd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( ( 𝑆 ↾ 𝐶 ) “ ( 𝐶 ∖ { 𝑋 } ) ) = ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) |
174 |
173
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐾 ‘ ∪ ( ( 𝑆 ↾ 𝐶 ) “ ( 𝐶 ∖ { 𝑋 } ) ) ) = ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) |
175 |
147 174
|
ineq12d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( ( 𝑆 ↾ 𝐶 ) “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) = ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ) |
176 |
148 149 67 5 10
|
dprddisj |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( ( 𝑆 ↾ 𝐶 ) “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) = { 0 } ) |
177 |
175 176
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) = { 0 } ) |
178 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
179 |
|
ffun |
⊢ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) → Fun 𝑆 ) |
180 |
|
funiunfv |
⊢ ( Fun 𝑆 → ∪ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ( 𝑆 ‘ 𝑦 ) = ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) |
181 |
178 179 180
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ( 𝑆 ‘ 𝑦 ) = ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) |
182 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → 𝐺 dom DProd ( 𝑆 ↾ 𝐶 ) ) |
183 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → dom ( 𝑆 ↾ 𝐶 ) = 𝐶 ) |
184 |
|
eldifi |
⊢ ( 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) → 𝑦 ∈ 𝐶 ) |
185 |
184
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → 𝑦 ∈ 𝐶 ) |
186 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → 𝑋 ∈ 𝐶 ) |
187 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) → 𝑦 ≠ 𝑋 ) |
188 |
187
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → 𝑦 ≠ 𝑋 ) |
189 |
182 183 185 186 188 4
|
dprdcntz |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑦 ) ⊆ ( 𝑍 ‘ ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) ) ) |
190 |
185
|
fvresd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑦 ) = ( 𝑆 ‘ 𝑦 ) ) |
191 |
25
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) = ( 𝑆 ‘ 𝑋 ) ) |
192 |
191
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → ( 𝑍 ‘ ( ( 𝑆 ↾ 𝐶 ) ‘ 𝑋 ) ) = ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
193 |
189 190 192
|
3sstr3d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) ∧ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ) → ( 𝑆 ‘ 𝑦 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
194 |
193
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∀ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ( 𝑆 ‘ 𝑦 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
195 |
|
iunss |
⊢ ( ∪ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ( 𝑆 ‘ 𝑦 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ↔ ∀ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ( 𝑆 ‘ 𝑦 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
196 |
194 195
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ 𝑦 ∈ ( 𝐶 ∖ { 𝑋 } ) ( 𝑆 ‘ 𝑦 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
197 |
181 196
|
eqsstrrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
198 |
39
|
subgss |
⊢ ( ( 𝑆 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( Base ‘ 𝐺 ) ) |
199 |
146 198
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( Base ‘ 𝐺 ) ) |
200 |
39 4
|
cntzsubg |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑆 ‘ 𝑋 ) ⊆ ( Base ‘ 𝐺 ) ) → ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
201 |
61 199 200
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
202 |
10
|
mrcsscl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
203 |
64 197 201 202
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑋 ) ) ) |
204 |
4 117 146 203
|
cntzrecd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ) ) |
205 |
121 146 117 120 5 170 177 4 204
|
lsmdisj3 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ‘ 𝑋 ) ∩ ( ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐶 ∖ { 𝑋 } ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ 𝐷 ) ) ) ) = { 0 } ) |
206 |
143 205
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ⊆ { 0 } ) |
207 |
58 206
|
jca |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝑌 ∈ 𝐼 → ( 𝑋 ≠ 𝑌 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑌 ) ) ) ) ∧ ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ⊆ { 0 } ) ) |