Step |
Hyp |
Ref |
Expression |
1 |
|
eldprdi.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
eldprdi.w |
⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 } |
3 |
|
eldprdi.1 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
4 |
|
eldprdi.2 |
⊢ ( 𝜑 → dom 𝑆 = 𝐼 ) |
5 |
|
eldprdi.3 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑊 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
7 |
2 3 4 5 6
|
dprdff |
⊢ ( 𝜑 → 𝐹 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
8 |
7
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) → 𝐹 = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
10 |
2 3 4 5
|
dprdfcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ) |
11 |
10
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ) |
12 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝐺 dom DProd 𝑆 ) |
13 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → dom 𝑆 = 𝐼 ) |
14 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝑥 ∈ 𝐼 ) |
15 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) |
16 |
1 2 12 13 14 11 15
|
dprdfid |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∈ 𝑊 ∧ ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
17 |
16
|
simpld |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∈ 𝑊 ) |
18 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝐹 ∈ 𝑊 ) |
19 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
20 |
1 2 12 13 17 18 19
|
dprdfsub |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∘f ( -g ‘ 𝐺 ) 𝐹 ) ∈ 𝑊 ∧ ( 𝐺 Σg ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∘f ( -g ‘ 𝐺 ) 𝐹 ) ) = ( ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ) ( -g ‘ 𝐺 ) ( 𝐺 Σg 𝐹 ) ) ) ) |
21 |
20
|
simprd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 Σg ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∘f ( -g ‘ 𝐺 ) 𝐹 ) ) = ( ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ) ( -g ‘ 𝐺 ) ( 𝐺 Σg 𝐹 ) ) ) |
22 |
3 4
|
dprddomcld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
23 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝐼 ∈ V ) |
24 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
25 |
1
|
fvexi |
⊢ 0 ∈ V |
26 |
24 25
|
ifex |
⊢ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ∈ V |
27 |
26
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) → if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ∈ V ) |
28 |
|
fvexd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑦 ) ∈ V ) |
29 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ) |
30 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝐹 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
31 |
30
|
feqmptd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝐹 = ( 𝑦 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑦 ) ) ) |
32 |
23 27 28 29 31
|
offval2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∘f ( -g ‘ 𝐺 ) 𝐹 ) = ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
33 |
32
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 Σg ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∘f ( -g ‘ 𝐺 ) 𝐹 ) ) = ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
34 |
16
|
simprd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
35 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 Σg 𝐹 ) = 0 ) |
36 |
34 35
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ) ( -g ‘ 𝐺 ) ( 𝐺 Σg 𝐹 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) 0 ) ) |
37 |
|
dprdgrp |
⊢ ( 𝐺 dom DProd 𝑆 → 𝐺 ∈ Grp ) |
38 |
12 37
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝐺 ∈ Grp ) |
39 |
30 14
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) ) |
40 |
6 1 19
|
grpsubid1 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) 0 ) = ( 𝐹 ‘ 𝑥 ) ) |
41 |
38 39 40
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) 0 ) = ( 𝐹 ‘ 𝑥 ) ) |
42 |
36 41
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ) ( -g ‘ 𝐺 ) ( 𝐺 Σg 𝐹 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
43 |
21 33 42
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
44 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
45 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
46 |
3 37 45
|
3syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
47 |
46
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝐺 ∈ Mnd ) |
48 |
6
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) ) |
49 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
50 |
38 48 49
|
3syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
51 |
|
imassrn |
⊢ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ ran 𝑆 |
52 |
3 4
|
dprdf2 |
⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
54 |
53
|
frnd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ran 𝑆 ⊆ ( SubGrp ‘ 𝐺 ) ) |
55 |
|
mresspw |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
56 |
50 55
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( SubGrp ‘ 𝐺 ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
57 |
54 56
|
sstrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ran 𝑆 ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
58 |
51 57
|
sstrid |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ) |
59 |
|
sspwuni |
⊢ ( ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ 𝒫 ( Base ‘ 𝐺 ) ↔ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐺 ) ) |
60 |
58 59
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐺 ) ) |
61 |
|
eqid |
⊢ ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
62 |
61
|
mrccl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ ( Base ‘ 𝐺 ) ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
63 |
50 60 62
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
64 |
|
subgsubm |
⊢ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubMnd ‘ 𝐺 ) ) |
65 |
63 64
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubMnd ‘ 𝐺 ) ) |
66 |
|
oveq1 |
⊢ ( ( 𝐹 ‘ 𝑥 ) = if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) = ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
67 |
66
|
eleq1d |
⊢ ( ( 𝐹 ‘ 𝑥 ) = if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) → ( ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ↔ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) ) |
68 |
|
oveq1 |
⊢ ( 0 = if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) → ( 0 ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) = ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
69 |
68
|
eleq1d |
⊢ ( 0 = if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) → ( ( 0 ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ↔ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) ) |
70 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑦 = 𝑥 ) → 𝑦 = 𝑥 ) |
71 |
70
|
fveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑦 = 𝑥 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑥 ) ) |
72 |
71
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑦 = 𝑥 ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑥 ) ) ) |
73 |
6 1 19
|
grpsubid |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑥 ) ) = 0 ) |
74 |
38 39 73
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑥 ) ) = 0 ) |
75 |
1
|
subg0cl |
⊢ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
76 |
63 75
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 0 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
77 |
74 76
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
78 |
77
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑦 = 𝑥 ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
79 |
72 78
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑦 = 𝑥 ) → ( ( 𝐹 ‘ 𝑥 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
80 |
63
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
81 |
80 75
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → 0 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
82 |
50 61 60
|
mrcssidd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
83 |
82
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
84 |
2 12 13 18
|
dprdfcl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑆 ‘ 𝑦 ) ) |
85 |
84
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑆 ‘ 𝑦 ) ) |
86 |
53
|
ffnd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → 𝑆 Fn 𝐼 ) |
87 |
86
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑆 Fn 𝐼 ) |
88 |
|
difssd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝐼 ∖ { 𝑥 } ) ⊆ 𝐼 ) |
89 |
|
df-ne |
⊢ ( 𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥 ) |
90 |
|
eldifsn |
⊢ ( 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ↔ ( 𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥 ) ) |
91 |
90
|
biimpri |
⊢ ( ( 𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥 ) → 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ) |
92 |
89 91
|
sylan2br |
⊢ ( ( 𝑦 ∈ 𝐼 ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ) |
93 |
92
|
adantll |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ) |
94 |
|
fnfvima |
⊢ ( ( 𝑆 Fn 𝐼 ∧ ( 𝐼 ∖ { 𝑥 } ) ⊆ 𝐼 ∧ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ) → ( 𝑆 ‘ 𝑦 ) ∈ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) |
95 |
87 88 93 94
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝑆 ‘ 𝑦 ) ∈ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) |
96 |
|
elunii |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑆 ‘ 𝑦 ) ∧ ( 𝑆 ‘ 𝑦 ) ∈ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) |
97 |
85 95 96
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ∈ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) |
98 |
83 97
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
99 |
19
|
subgsubcl |
⊢ ( ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ 0 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) → ( 0 ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
100 |
80 81 98 99
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 0 ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
101 |
67 69 79 100
|
ifbothda |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑦 ∈ 𝐼 ) → ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
102 |
101
|
fmpttd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) : 𝐼 ⟶ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
103 |
20
|
simpld |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ) ∘f ( -g ‘ 𝐺 ) 𝐹 ) ∈ 𝑊 ) |
104 |
32 103
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ∈ 𝑊 ) |
105 |
2 12 13 104 44
|
dprdfcntz |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ran ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
106 |
2 12 13 104
|
dprdffsupp |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) finSupp 0 ) |
107 |
1 44 47 23 65 102 105 106
|
gsumzsubmcl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 Σg ( 𝑦 ∈ 𝐼 ↦ ( if ( 𝑦 = 𝑥 , ( 𝐹 ‘ 𝑥 ) , 0 ) ( -g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
108 |
43 107
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
109 |
11 108
|
elind |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) ) |
110 |
12 13 14 1 61
|
dprddisj |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑆 ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) |
111 |
109 110
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ { 0 } ) |
112 |
|
elsni |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ { 0 } → ( 𝐹 ‘ 𝑥 ) = 0 ) |
113 |
111 112
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
114 |
113
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) → ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐼 ↦ 0 ) ) |
115 |
9 114
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝐺 Σg 𝐹 ) = 0 ) → 𝐹 = ( 𝑥 ∈ 𝐼 ↦ 0 ) ) |
116 |
115
|
ex |
⊢ ( 𝜑 → ( ( 𝐺 Σg 𝐹 ) = 0 → 𝐹 = ( 𝑥 ∈ 𝐼 ↦ 0 ) ) ) |
117 |
1
|
gsumz |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐼 ∈ V ) → ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 0 ) ) = 0 ) |
118 |
46 22 117
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 0 ) ) = 0 ) |
119 |
|
oveq2 |
⊢ ( 𝐹 = ( 𝑥 ∈ 𝐼 ↦ 0 ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 0 ) ) ) |
120 |
119
|
eqeq1d |
⊢ ( 𝐹 = ( 𝑥 ∈ 𝐼 ↦ 0 ) → ( ( 𝐺 Σg 𝐹 ) = 0 ↔ ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 0 ) ) = 0 ) ) |
121 |
118 120
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝐹 = ( 𝑥 ∈ 𝐼 ↦ 0 ) → ( 𝐺 Σg 𝐹 ) = 0 ) ) |
122 |
116 121
|
impbid |
⊢ ( 𝜑 → ( ( 𝐺 Σg 𝐹 ) = 0 ↔ 𝐹 = ( 𝑥 ∈ 𝐼 ↦ 0 ) ) ) |