Step |
Hyp |
Ref |
Expression |
1 |
|
dpjfval.1 |
|- ( ph -> G dom DProd S ) |
2 |
|
dpjfval.2 |
|- ( ph -> dom S = I ) |
3 |
|
dpjfval.p |
|- P = ( G dProj S ) |
4 |
|
dpjidcl.3 |
|- ( ph -> A e. ( G DProd S ) ) |
5 |
|
dpjidcl.0 |
|- .0. = ( 0g ` G ) |
6 |
|
dpjidcl.w |
|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. } |
7 |
5 6
|
eldprd |
|- ( dom S = I -> ( A e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W A = ( G gsum f ) ) ) ) |
8 |
2 7
|
syl |
|- ( ph -> ( A e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W A = ( G gsum f ) ) ) ) |
9 |
4 8
|
mpbid |
|- ( ph -> ( G dom DProd S /\ E. f e. W A = ( G gsum f ) ) ) |
10 |
9
|
simprd |
|- ( ph -> E. f e. W A = ( G gsum f ) ) |
11 |
1
|
adantr |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> G dom DProd S ) |
12 |
2
|
adantr |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> dom S = I ) |
13 |
1
|
ad2antrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> G dom DProd S ) |
14 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> dom S = I ) |
15 |
|
simpr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> x e. I ) |
16 |
13 14 3 15
|
dpjf |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( P ` x ) : ( G DProd S ) --> ( S ` x ) ) |
17 |
4
|
ad2antrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> A e. ( G DProd S ) ) |
18 |
16 17
|
ffvelrnd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) e. ( S ` x ) ) |
19 |
1 2
|
dprddomcld |
|- ( ph -> I e. _V ) |
20 |
19
|
mptexd |
|- ( ph -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V ) |
21 |
20
|
adantr |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V ) |
22 |
|
funmpt |
|- Fun ( x e. I |-> ( ( P ` x ) ` A ) ) |
23 |
22
|
a1i |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> Fun ( x e. I |-> ( ( P ` x ) ` A ) ) ) |
24 |
|
simprl |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> f e. W ) |
25 |
6 11 12 24
|
dprdffsupp |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> f finSupp .0. ) |
26 |
|
eldifi |
|- ( x e. ( I \ ( f supp .0. ) ) -> x e. I ) |
27 |
|
eqid |
|- ( proj1 ` G ) = ( proj1 ` G ) |
28 |
13 14 3 27 15
|
dpjval |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( P ` x ) = ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ) |
29 |
28
|
fveq1d |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) = ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) ) |
30 |
26 29
|
sylan2 |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( ( P ` x ) ` A ) = ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) ) |
31 |
|
simplrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A = ( G gsum f ) ) |
32 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
33 |
|
eqid |
|- ( Cntz ` G ) = ( Cntz ` G ) |
34 |
|
dprdgrp |
|- ( G dom DProd S -> G e. Grp ) |
35 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
36 |
11 34 35
|
3syl |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> G e. Mnd ) |
37 |
36
|
adantr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> G e. Mnd ) |
38 |
19
|
ad2antrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> I e. _V ) |
39 |
6 11 12 24 32
|
dprdff |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> f : I --> ( Base ` G ) ) |
40 |
39
|
adantr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> f : I --> ( Base ` G ) ) |
41 |
24
|
adantr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> f e. W ) |
42 |
6 13 14 41 33
|
dprdfcntz |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ran f C_ ( ( Cntz ` G ) ` ran f ) ) |
43 |
26 42
|
sylan2 |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ran f C_ ( ( Cntz ` G ) ` ran f ) ) |
44 |
|
snssi |
|- ( x e. ( I \ ( f supp .0. ) ) -> { x } C_ ( I \ ( f supp .0. ) ) ) |
45 |
44
|
adantl |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> { x } C_ ( I \ ( f supp .0. ) ) ) |
46 |
45
|
difss2d |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> { x } C_ I ) |
47 |
|
suppssdm |
|- ( f supp .0. ) C_ dom f |
48 |
47 39
|
fssdm |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( f supp .0. ) C_ I ) |
49 |
48
|
adantr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( f supp .0. ) C_ I ) |
50 |
|
ssconb |
|- ( ( { x } C_ I /\ ( f supp .0. ) C_ I ) -> ( { x } C_ ( I \ ( f supp .0. ) ) <-> ( f supp .0. ) C_ ( I \ { x } ) ) ) |
51 |
46 49 50
|
syl2anc |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( { x } C_ ( I \ ( f supp .0. ) ) <-> ( f supp .0. ) C_ ( I \ { x } ) ) ) |
52 |
45 51
|
mpbid |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( f supp .0. ) C_ ( I \ { x } ) ) |
53 |
25
|
adantr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> f finSupp .0. ) |
54 |
32 5 33 37 38 40 43 52 53
|
gsumzres |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( G gsum ( f |` ( I \ { x } ) ) ) = ( G gsum f ) ) |
55 |
31 54
|
eqtr4d |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A = ( G gsum ( f |` ( I \ { x } ) ) ) ) |
56 |
|
eqid |
|- { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. } = { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. } |
57 |
|
difss |
|- ( I \ { x } ) C_ I |
58 |
57
|
a1i |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( I \ { x } ) C_ I ) |
59 |
13 14 58
|
dprdres |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G dom DProd ( S |` ( I \ { x } ) ) /\ ( G DProd ( S |` ( I \ { x } ) ) ) C_ ( G DProd S ) ) ) |
60 |
59
|
simpld |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> G dom DProd ( S |` ( I \ { x } ) ) ) |
61 |
13 14
|
dprdf2 |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> S : I --> ( SubGrp ` G ) ) |
62 |
|
fssres |
|- ( ( S : I --> ( SubGrp ` G ) /\ ( I \ { x } ) C_ I ) -> ( S |` ( I \ { x } ) ) : ( I \ { x } ) --> ( SubGrp ` G ) ) |
63 |
61 57 62
|
sylancl |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( S |` ( I \ { x } ) ) : ( I \ { x } ) --> ( SubGrp ` G ) ) |
64 |
63
|
fdmd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> dom ( S |` ( I \ { x } ) ) = ( I \ { x } ) ) |
65 |
39
|
adantr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> f : I --> ( Base ` G ) ) |
66 |
65
|
feqmptd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> f = ( k e. I |-> ( f ` k ) ) ) |
67 |
66
|
reseq1d |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) = ( ( k e. I |-> ( f ` k ) ) |` ( I \ { x } ) ) ) |
68 |
|
resmpt |
|- ( ( I \ { x } ) C_ I -> ( ( k e. I |-> ( f ` k ) ) |` ( I \ { x } ) ) = ( k e. ( I \ { x } ) |-> ( f ` k ) ) ) |
69 |
57 68
|
ax-mp |
|- ( ( k e. I |-> ( f ` k ) ) |` ( I \ { x } ) ) = ( k e. ( I \ { x } ) |-> ( f ` k ) ) |
70 |
67 69
|
eqtrdi |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) = ( k e. ( I \ { x } ) |-> ( f ` k ) ) ) |
71 |
|
eldifi |
|- ( k e. ( I \ { x } ) -> k e. I ) |
72 |
6 13 14 41
|
dprdfcl |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) /\ k e. I ) -> ( f ` k ) e. ( S ` k ) ) |
73 |
71 72
|
sylan2 |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) /\ k e. ( I \ { x } ) ) -> ( f ` k ) e. ( S ` k ) ) |
74 |
|
fvres |
|- ( k e. ( I \ { x } ) -> ( ( S |` ( I \ { x } ) ) ` k ) = ( S ` k ) ) |
75 |
74
|
adantl |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) /\ k e. ( I \ { x } ) ) -> ( ( S |` ( I \ { x } ) ) ` k ) = ( S ` k ) ) |
76 |
73 75
|
eleqtrrd |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) /\ k e. ( I \ { x } ) ) -> ( f ` k ) e. ( ( S |` ( I \ { x } ) ) ` k ) ) |
77 |
|
difexg |
|- ( I e. _V -> ( I \ { x } ) e. _V ) |
78 |
19 77
|
syl |
|- ( ph -> ( I \ { x } ) e. _V ) |
79 |
78
|
mptexd |
|- ( ph -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V ) |
80 |
79
|
ad2antrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V ) |
81 |
|
funmpt |
|- Fun ( k e. ( I \ { x } ) |-> ( f ` k ) ) |
82 |
81
|
a1i |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> Fun ( k e. ( I \ { x } ) |-> ( f ` k ) ) ) |
83 |
25
|
adantr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> f finSupp .0. ) |
84 |
|
ssdif |
|- ( ( I \ { x } ) C_ I -> ( ( I \ { x } ) \ ( f supp .0. ) ) C_ ( I \ ( f supp .0. ) ) ) |
85 |
57 84
|
ax-mp |
|- ( ( I \ { x } ) \ ( f supp .0. ) ) C_ ( I \ ( f supp .0. ) ) |
86 |
85
|
sseli |
|- ( k e. ( ( I \ { x } ) \ ( f supp .0. ) ) -> k e. ( I \ ( f supp .0. ) ) ) |
87 |
|
ssidd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( f supp .0. ) C_ ( f supp .0. ) ) |
88 |
19
|
ad2antrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> I e. _V ) |
89 |
5
|
fvexi |
|- .0. e. _V |
90 |
89
|
a1i |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> .0. e. _V ) |
91 |
65 87 88 90
|
suppssr |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) /\ k e. ( I \ ( f supp .0. ) ) ) -> ( f ` k ) = .0. ) |
92 |
86 91
|
sylan2 |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) /\ k e. ( ( I \ { x } ) \ ( f supp .0. ) ) ) -> ( f ` k ) = .0. ) |
93 |
78
|
ad2antrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( I \ { x } ) e. _V ) |
94 |
92 93
|
suppss2 |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( k e. ( I \ { x } ) |-> ( f ` k ) ) supp .0. ) C_ ( f supp .0. ) ) |
95 |
|
fsuppsssupp |
|- ( ( ( ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V /\ Fun ( k e. ( I \ { x } ) |-> ( f ` k ) ) ) /\ ( f finSupp .0. /\ ( ( k e. ( I \ { x } ) |-> ( f ` k ) ) supp .0. ) C_ ( f supp .0. ) ) ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) finSupp .0. ) |
96 |
80 82 83 94 95
|
syl22anc |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) finSupp .0. ) |
97 |
56 60 64 76 96
|
dprdwd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. } ) |
98 |
70 97
|
eqeltrd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) e. { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. } ) |
99 |
5 56 60 64 98
|
eldprdi |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G gsum ( f |` ( I \ { x } ) ) ) e. ( G DProd ( S |` ( I \ { x } ) ) ) ) |
100 |
26 99
|
sylan2 |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( G gsum ( f |` ( I \ { x } ) ) ) e. ( G DProd ( S |` ( I \ { x } ) ) ) ) |
101 |
55 100
|
eqeltrd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A e. ( G DProd ( S |` ( I \ { x } ) ) ) ) |
102 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
103 |
|
eqid |
|- ( LSSum ` G ) = ( LSSum ` G ) |
104 |
61 15
|
ffvelrnd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( S ` x ) e. ( SubGrp ` G ) ) |
105 |
|
dprdsubg |
|- ( G dom DProd ( S |` ( I \ { x } ) ) -> ( G DProd ( S |` ( I \ { x } ) ) ) e. ( SubGrp ` G ) ) |
106 |
60 105
|
syl |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G DProd ( S |` ( I \ { x } ) ) ) e. ( SubGrp ` G ) ) |
107 |
13 14 15 5
|
dpjdisj |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( S ` x ) i^i ( G DProd ( S |` ( I \ { x } ) ) ) ) = { .0. } ) |
108 |
13 14 15 33
|
dpjcntz |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( S ` x ) C_ ( ( Cntz ` G ) ` ( G DProd ( S |` ( I \ { x } ) ) ) ) ) |
109 |
102 103 5 33 104 106 107 108 27
|
pj1rid |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) /\ A e. ( G DProd ( S |` ( I \ { x } ) ) ) ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = .0. ) |
110 |
26 109
|
sylanl2 |
|- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) /\ A e. ( G DProd ( S |` ( I \ { x } ) ) ) ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = .0. ) |
111 |
101 110
|
mpdan |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = .0. ) |
112 |
30 111
|
eqtrd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( ( P ` x ) ` A ) = .0. ) |
113 |
19
|
adantr |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> I e. _V ) |
114 |
112 113
|
suppss2 |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) supp .0. ) C_ ( f supp .0. ) ) |
115 |
|
fsuppsssupp |
|- ( ( ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V /\ Fun ( x e. I |-> ( ( P ` x ) ` A ) ) ) /\ ( f finSupp .0. /\ ( ( x e. I |-> ( ( P ` x ) ` A ) ) supp .0. ) C_ ( f supp .0. ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) finSupp .0. ) |
116 |
21 23 25 114 115
|
syl22anc |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) finSupp .0. ) |
117 |
6 11 12 18 116
|
dprdwd |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. W ) |
118 |
|
simprr |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> A = ( G gsum f ) ) |
119 |
39
|
feqmptd |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> f = ( x e. I |-> ( f ` x ) ) ) |
120 |
|
simplrr |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> A = ( G gsum f ) ) |
121 |
13 34 35
|
3syl |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> G e. Mnd ) |
122 |
6 13 14 41
|
dprdffsupp |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> f finSupp .0. ) |
123 |
|
disjdif |
|- ( { x } i^i ( I \ { x } ) ) = (/) |
124 |
123
|
a1i |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( { x } i^i ( I \ { x } ) ) = (/) ) |
125 |
|
undif2 |
|- ( { x } u. ( I \ { x } ) ) = ( { x } u. I ) |
126 |
15
|
snssd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> { x } C_ I ) |
127 |
|
ssequn1 |
|- ( { x } C_ I <-> ( { x } u. I ) = I ) |
128 |
126 127
|
sylib |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( { x } u. I ) = I ) |
129 |
125 128
|
syl5req |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> I = ( { x } u. ( I \ { x } ) ) ) |
130 |
32 5 102 33 121 88 65 42 122 124 129
|
gsumzsplit |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G gsum f ) = ( ( G gsum ( f |` { x } ) ) ( +g ` G ) ( G gsum ( f |` ( I \ { x } ) ) ) ) ) |
131 |
65 126
|
feqresmpt |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( f |` { x } ) = ( k e. { x } |-> ( f ` k ) ) ) |
132 |
131
|
oveq2d |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G gsum ( f |` { x } ) ) = ( G gsum ( k e. { x } |-> ( f ` k ) ) ) ) |
133 |
65 15
|
ffvelrnd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( f ` x ) e. ( Base ` G ) ) |
134 |
|
fveq2 |
|- ( k = x -> ( f ` k ) = ( f ` x ) ) |
135 |
32 134
|
gsumsn |
|- ( ( G e. Mnd /\ x e. I /\ ( f ` x ) e. ( Base ` G ) ) -> ( G gsum ( k e. { x } |-> ( f ` k ) ) ) = ( f ` x ) ) |
136 |
121 15 133 135
|
syl3anc |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G gsum ( k e. { x } |-> ( f ` k ) ) ) = ( f ` x ) ) |
137 |
132 136
|
eqtrd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G gsum ( f |` { x } ) ) = ( f ` x ) ) |
138 |
137
|
oveq1d |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( G gsum ( f |` { x } ) ) ( +g ` G ) ( G gsum ( f |` ( I \ { x } ) ) ) ) = ( ( f ` x ) ( +g ` G ) ( G gsum ( f |` ( I \ { x } ) ) ) ) ) |
139 |
120 130 138
|
3eqtrd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> A = ( ( f ` x ) ( +g ` G ) ( G gsum ( f |` ( I \ { x } ) ) ) ) ) |
140 |
13 14 15 103
|
dpjlsm |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( G DProd S ) = ( ( S ` x ) ( LSSum ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ) |
141 |
17 140
|
eleqtrd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> A e. ( ( S ` x ) ( LSSum ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ) |
142 |
6 11 12 24
|
dprdfcl |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( f ` x ) e. ( S ` x ) ) |
143 |
102 103 5 33 104 106 107 108 27 141 142 99
|
pj1eq |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( A = ( ( f ` x ) ( +g ` G ) ( G gsum ( f |` ( I \ { x } ) ) ) ) <-> ( ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = ( f ` x ) /\ ( ( ( G DProd ( S |` ( I \ { x } ) ) ) ( proj1 ` G ) ( S ` x ) ) ` A ) = ( G gsum ( f |` ( I \ { x } ) ) ) ) ) ) |
144 |
139 143
|
mpbid |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = ( f ` x ) /\ ( ( ( G DProd ( S |` ( I \ { x } ) ) ) ( proj1 ` G ) ( S ` x ) ) ` A ) = ( G gsum ( f |` ( I \ { x } ) ) ) ) ) |
145 |
144
|
simpld |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = ( f ` x ) ) |
146 |
29 145
|
eqtrd |
|- ( ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) = ( f ` x ) ) |
147 |
146
|
mpteq2dva |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) = ( x e. I |-> ( f ` x ) ) ) |
148 |
119 147
|
eqtr4d |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> f = ( x e. I |-> ( ( P ` x ) ` A ) ) ) |
149 |
148
|
oveq2d |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( G gsum f ) = ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) |
150 |
118 149
|
eqtrd |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> A = ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) |
151 |
117 150
|
jca |
|- ( ( ph /\ ( f e. W /\ A = ( G gsum f ) ) ) -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. W /\ A = ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) ) |
152 |
10 151
|
rexlimddv |
|- ( ph -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. W /\ A = ( G gsum ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) ) |