Step |
Hyp |
Ref |
Expression |
1 |
|
dmdprdsplitlem.0 |
|- .0. = ( 0g ` G ) |
2 |
|
dmdprdsplitlem.w |
|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. } |
3 |
|
dmdprdsplitlem.1 |
|- ( ph -> G dom DProd S ) |
4 |
|
dmdprdsplitlem.2 |
|- ( ph -> dom S = I ) |
5 |
|
dmdprdsplitlem.3 |
|- ( ph -> A C_ I ) |
6 |
|
dmdprdsplitlem.4 |
|- ( ph -> F e. W ) |
7 |
|
dmdprdsplitlem.5 |
|- ( ph -> ( G gsum F ) e. ( G DProd ( S |` A ) ) ) |
8 |
3 4
|
dprdf2 |
|- ( ph -> S : I --> ( SubGrp ` G ) ) |
9 |
8 5
|
fssresd |
|- ( ph -> ( S |` A ) : A --> ( SubGrp ` G ) ) |
10 |
|
fdm |
|- ( ( S |` A ) : A --> ( SubGrp ` G ) -> dom ( S |` A ) = A ) |
11 |
|
eqid |
|- { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } = { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } |
12 |
1 11
|
eldprd |
|- ( dom ( S |` A ) = A -> ( ( G gsum F ) e. ( G DProd ( S |` A ) ) <-> ( G dom DProd ( S |` A ) /\ E. f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } ( G gsum F ) = ( G gsum f ) ) ) ) |
13 |
9 10 12
|
3syl |
|- ( ph -> ( ( G gsum F ) e. ( G DProd ( S |` A ) ) <-> ( G dom DProd ( S |` A ) /\ E. f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } ( G gsum F ) = ( G gsum f ) ) ) ) |
14 |
7 13
|
mpbid |
|- ( ph -> ( G dom DProd ( S |` A ) /\ E. f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } ( G gsum F ) = ( G gsum f ) ) ) |
15 |
14
|
simprd |
|- ( ph -> E. f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } ( G gsum F ) = ( G gsum f ) ) |
16 |
15
|
adantr |
|- ( ( ph /\ X e. ( I \ A ) ) -> E. f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } ( G gsum F ) = ( G gsum f ) ) |
17 |
|
simprr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( G gsum F ) = ( G gsum f ) ) |
18 |
14
|
simpld |
|- ( ph -> G dom DProd ( S |` A ) ) |
19 |
18
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> G dom DProd ( S |` A ) ) |
20 |
9 10
|
syl |
|- ( ph -> dom ( S |` A ) = A ) |
21 |
20
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> dom ( S |` A ) = A ) |
22 |
|
simprl |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } ) |
23 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
24 |
11 19 21 22 23
|
dprdff |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> f : A --> ( Base ` G ) ) |
25 |
24
|
feqmptd |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> f = ( n e. A |-> ( f ` n ) ) ) |
26 |
5
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> A C_ I ) |
27 |
26
|
resmptd |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) |` A ) = ( n e. A |-> if ( n e. A , ( f ` n ) , .0. ) ) ) |
28 |
|
iftrue |
|- ( n e. A -> if ( n e. A , ( f ` n ) , .0. ) = ( f ` n ) ) |
29 |
28
|
mpteq2ia |
|- ( n e. A |-> if ( n e. A , ( f ` n ) , .0. ) ) = ( n e. A |-> ( f ` n ) ) |
30 |
27 29
|
eqtrdi |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) |` A ) = ( n e. A |-> ( f ` n ) ) ) |
31 |
25 30
|
eqtr4d |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> f = ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) |` A ) ) |
32 |
31
|
oveq2d |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( G gsum f ) = ( G gsum ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) |` A ) ) ) |
33 |
|
eqid |
|- ( Cntz ` G ) = ( Cntz ` G ) |
34 |
3
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> G dom DProd S ) |
35 |
|
dprdgrp |
|- ( G dom DProd S -> G e. Grp ) |
36 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
37 |
34 35 36
|
3syl |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> G e. Mnd ) |
38 |
3 4
|
dprddomcld |
|- ( ph -> I e. _V ) |
39 |
38
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> I e. _V ) |
40 |
4
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> dom S = I ) |
41 |
19
|
adantr |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) -> G dom DProd ( S |` A ) ) |
42 |
21
|
adantr |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) -> dom ( S |` A ) = A ) |
43 |
|
simplrl |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) -> f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } ) |
44 |
11 41 42 43
|
dprdfcl |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) /\ n e. A ) -> ( f ` n ) e. ( ( S |` A ) ` n ) ) |
45 |
|
fvres |
|- ( n e. A -> ( ( S |` A ) ` n ) = ( S ` n ) ) |
46 |
45
|
adantl |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) /\ n e. A ) -> ( ( S |` A ) ` n ) = ( S ` n ) ) |
47 |
44 46
|
eleqtrd |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) /\ n e. A ) -> ( f ` n ) e. ( S ` n ) ) |
48 |
8
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> S : I --> ( SubGrp ` G ) ) |
49 |
48
|
ffvelrnda |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) -> ( S ` n ) e. ( SubGrp ` G ) ) |
50 |
1
|
subg0cl |
|- ( ( S ` n ) e. ( SubGrp ` G ) -> .0. e. ( S ` n ) ) |
51 |
49 50
|
syl |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) -> .0. e. ( S ` n ) ) |
52 |
51
|
adantr |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) /\ -. n e. A ) -> .0. e. ( S ` n ) ) |
53 |
47 52
|
ifclda |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. I ) -> if ( n e. A , ( f ` n ) , .0. ) e. ( S ` n ) ) |
54 |
38
|
mptexd |
|- ( ph -> ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) e. _V ) |
55 |
54
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) e. _V ) |
56 |
|
funmpt |
|- Fun ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) |
57 |
56
|
a1i |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> Fun ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) |
58 |
11 19 21 22
|
dprdffsupp |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> f finSupp .0. ) |
59 |
|
simpr |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ ( f supp .0. ) ) ) /\ n e. A ) -> n e. A ) |
60 |
|
eldifn |
|- ( n e. ( I \ ( f supp .0. ) ) -> -. n e. ( f supp .0. ) ) |
61 |
60
|
ad2antlr |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ ( f supp .0. ) ) ) /\ n e. A ) -> -. n e. ( f supp .0. ) ) |
62 |
59 61
|
eldifd |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ ( f supp .0. ) ) ) /\ n e. A ) -> n e. ( A \ ( f supp .0. ) ) ) |
63 |
|
ssidd |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( f supp .0. ) C_ ( f supp .0. ) ) |
64 |
38 5
|
ssexd |
|- ( ph -> A e. _V ) |
65 |
64
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> A e. _V ) |
66 |
1
|
fvexi |
|- .0. e. _V |
67 |
66
|
a1i |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> .0. e. _V ) |
68 |
24 63 65 67
|
suppssr |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( A \ ( f supp .0. ) ) ) -> ( f ` n ) = .0. ) |
69 |
68
|
adantlr |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ ( f supp .0. ) ) ) /\ n e. ( A \ ( f supp .0. ) ) ) -> ( f ` n ) = .0. ) |
70 |
62 69
|
syldan |
|- ( ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ ( f supp .0. ) ) ) /\ n e. A ) -> ( f ` n ) = .0. ) |
71 |
70
|
ifeq1da |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ ( f supp .0. ) ) ) -> if ( n e. A , ( f ` n ) , .0. ) = if ( n e. A , .0. , .0. ) ) |
72 |
|
ifid |
|- if ( n e. A , .0. , .0. ) = .0. |
73 |
71 72
|
eqtrdi |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ ( f supp .0. ) ) ) -> if ( n e. A , ( f ` n ) , .0. ) = .0. ) |
74 |
73 39
|
suppss2 |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) supp .0. ) C_ ( f supp .0. ) ) |
75 |
|
fsuppsssupp |
|- ( ( ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) e. _V /\ Fun ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) /\ ( f finSupp .0. /\ ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) supp .0. ) C_ ( f supp .0. ) ) ) -> ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) finSupp .0. ) |
76 |
55 57 58 74 75
|
syl22anc |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) finSupp .0. ) |
77 |
2 34 40 53 76
|
dprdwd |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) e. W ) |
78 |
2 34 40 77 23
|
dprdff |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) : I --> ( Base ` G ) ) |
79 |
2 34 40 77 33
|
dprdfcntz |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ran ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) C_ ( ( Cntz ` G ) ` ran ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) ) |
80 |
|
eldifn |
|- ( n e. ( I \ A ) -> -. n e. A ) |
81 |
80
|
adantl |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ A ) ) -> -. n e. A ) |
82 |
81
|
iffalsed |
|- ( ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) /\ n e. ( I \ A ) ) -> if ( n e. A , ( f ` n ) , .0. ) = .0. ) |
83 |
82 39
|
suppss2 |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) supp .0. ) C_ A ) |
84 |
23 1 33 37 39 78 79 83 76
|
gsumzres |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( G gsum ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) |` A ) ) = ( G gsum ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) ) |
85 |
17 32 84
|
3eqtrd |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( G gsum F ) = ( G gsum ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) ) |
86 |
6
|
ad2antrr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> F e. W ) |
87 |
1 2 34 40 86 77
|
dprdf11 |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( ( G gsum F ) = ( G gsum ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) <-> F = ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) ) |
88 |
85 87
|
mpbid |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> F = ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ) |
89 |
88
|
fveq1d |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( F ` X ) = ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ` X ) ) |
90 |
|
eldifi |
|- ( X e. ( I \ A ) -> X e. I ) |
91 |
90
|
ad2antlr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> X e. I ) |
92 |
|
eleq1 |
|- ( n = X -> ( n e. A <-> X e. A ) ) |
93 |
|
fveq2 |
|- ( n = X -> ( f ` n ) = ( f ` X ) ) |
94 |
92 93
|
ifbieq1d |
|- ( n = X -> if ( n e. A , ( f ` n ) , .0. ) = if ( X e. A , ( f ` X ) , .0. ) ) |
95 |
|
eqid |
|- ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) = ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) |
96 |
|
fvex |
|- ( f ` n ) e. _V |
97 |
96 66
|
ifex |
|- if ( n e. A , ( f ` n ) , .0. ) e. _V |
98 |
94 95 97
|
fvmpt3i |
|- ( X e. I -> ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ` X ) = if ( X e. A , ( f ` X ) , .0. ) ) |
99 |
91 98
|
syl |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( ( n e. I |-> if ( n e. A , ( f ` n ) , .0. ) ) ` X ) = if ( X e. A , ( f ` X ) , .0. ) ) |
100 |
|
eldifn |
|- ( X e. ( I \ A ) -> -. X e. A ) |
101 |
100
|
ad2antlr |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> -. X e. A ) |
102 |
101
|
iffalsed |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> if ( X e. A , ( f ` X ) , .0. ) = .0. ) |
103 |
89 99 102
|
3eqtrd |
|- ( ( ( ph /\ X e. ( I \ A ) ) /\ ( f e. { h e. X_ i e. A ( ( S |` A ) ` i ) | h finSupp .0. } /\ ( G gsum F ) = ( G gsum f ) ) ) -> ( F ` X ) = .0. ) |
104 |
16 103
|
rexlimddv |
|- ( ( ph /\ X e. ( I \ A ) ) -> ( F ` X ) = .0. ) |