| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsumfs2d.p |
|- F/ x ph |
| 2 |
|
gsumfs2d.b |
|- B = ( Base ` W ) |
| 3 |
|
gsumfs2d.1 |
|- .0. = ( 0g ` W ) |
| 4 |
|
gsumfs2d.r |
|- ( ph -> Rel A ) |
| 5 |
|
gsumfs2d.2 |
|- ( ph -> F finSupp .0. ) |
| 6 |
|
gsumfs2d.w |
|- ( ph -> W e. CMnd ) |
| 7 |
|
gsumfs2d.3 |
|- ( ph -> F : A --> B ) |
| 8 |
|
gsumfs2d.a |
|- ( ph -> A e. X ) |
| 9 |
6
|
adantr |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> W e. CMnd ) |
| 10 |
8
|
adantr |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> A e. X ) |
| 11 |
10
|
imaexd |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> ( A " { x } ) e. _V ) |
| 12 |
7
|
ffnd |
|- ( ph -> F Fn A ) |
| 13 |
12
|
ad2antrr |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> F Fn A ) |
| 14 |
8
|
ad2antrr |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> A e. X ) |
| 15 |
3
|
fvexi |
|- .0. e. _V |
| 16 |
15
|
a1i |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> .0. e. _V ) |
| 17 |
|
simpr |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) |
| 18 |
17
|
eldifad |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> y e. ( A " { x } ) ) |
| 19 |
|
vex |
|- x e. _V |
| 20 |
|
vex |
|- y e. _V |
| 21 |
19 20
|
elimasn |
|- ( y e. ( A " { x } ) <-> <. x , y >. e. A ) |
| 22 |
21
|
biimpi |
|- ( y e. ( A " { x } ) -> <. x , y >. e. A ) |
| 23 |
18 22
|
syl |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> <. x , y >. e. A ) |
| 24 |
17
|
eldifbd |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> -. y e. ( ( F supp .0. ) " { x } ) ) |
| 25 |
19 20
|
elimasn |
|- ( y e. ( ( F supp .0. ) " { x } ) <-> <. x , y >. e. ( F supp .0. ) ) |
| 26 |
25
|
biimpri |
|- ( <. x , y >. e. ( F supp .0. ) -> y e. ( ( F supp .0. ) " { x } ) ) |
| 27 |
24 26
|
nsyl |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> -. <. x , y >. e. ( F supp .0. ) ) |
| 28 |
23 27
|
eldifd |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> <. x , y >. e. ( A \ ( F supp .0. ) ) ) |
| 29 |
13 14 16 28
|
fvdifsupp |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( ( A " { x } ) \ ( ( F supp .0. ) " { x } ) ) ) -> ( F ` <. x , y >. ) = .0. ) |
| 30 |
5
|
fsuppimpd |
|- ( ph -> ( F supp .0. ) e. Fin ) |
| 31 |
30
|
adantr |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> ( F supp .0. ) e. Fin ) |
| 32 |
|
imafi2 |
|- ( ( F supp .0. ) e. Fin -> ( ( F supp .0. ) " { x } ) e. Fin ) |
| 33 |
31 32
|
syl |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> ( ( F supp .0. ) " { x } ) e. Fin ) |
| 34 |
7
|
ad2antrr |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( A " { x } ) ) -> F : A --> B ) |
| 35 |
22
|
adantl |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( A " { x } ) ) -> <. x , y >. e. A ) |
| 36 |
34 35
|
ffvelcdmd |
|- ( ( ( ph /\ x e. dom ( F supp .0. ) ) /\ y e. ( A " { x } ) ) -> ( F ` <. x , y >. ) e. B ) |
| 37 |
|
suppssdm |
|- ( F supp .0. ) C_ dom F |
| 38 |
37 7
|
fssdm |
|- ( ph -> ( F supp .0. ) C_ A ) |
| 39 |
38
|
adantr |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> ( F supp .0. ) C_ A ) |
| 40 |
|
imass1 |
|- ( ( F supp .0. ) C_ A -> ( ( F supp .0. ) " { x } ) C_ ( A " { x } ) ) |
| 41 |
39 40
|
syl |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> ( ( F supp .0. ) " { x } ) C_ ( A " { x } ) ) |
| 42 |
2 3 9 11 29 33 36 41
|
gsummptres2 |
|- ( ( ph /\ x e. dom ( F supp .0. ) ) -> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) = ( W gsum ( y e. ( ( F supp .0. ) " { x } ) |-> ( F ` <. x , y >. ) ) ) ) |
| 43 |
42
|
mpteq2dva |
|- ( ph -> ( x e. dom ( F supp .0. ) |-> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) ) = ( x e. dom ( F supp .0. ) |-> ( W gsum ( y e. ( ( F supp .0. ) " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) |
| 44 |
43
|
oveq2d |
|- ( ph -> ( W gsum ( x e. dom ( F supp .0. ) |-> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) = ( W gsum ( x e. dom ( F supp .0. ) |-> ( W gsum ( y e. ( ( F supp .0. ) " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) ) |
| 45 |
8
|
dmexd |
|- ( ph -> dom A e. _V ) |
| 46 |
12
|
ad2antrr |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> F Fn A ) |
| 47 |
8
|
ad2antrr |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> A e. X ) |
| 48 |
15
|
a1i |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> .0. e. _V ) |
| 49 |
22
|
adantl |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> <. x , y >. e. A ) |
| 50 |
|
simplr |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> x e. ( dom A \ dom ( F supp .0. ) ) ) |
| 51 |
50
|
eldifbd |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> -. x e. dom ( F supp .0. ) ) |
| 52 |
19 20
|
opeldm |
|- ( <. x , y >. e. ( F supp .0. ) -> x e. dom ( F supp .0. ) ) |
| 53 |
51 52
|
nsyl |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> -. <. x , y >. e. ( F supp .0. ) ) |
| 54 |
49 53
|
eldifd |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> <. x , y >. e. ( A \ ( F supp .0. ) ) ) |
| 55 |
46 47 48 54
|
fvdifsupp |
|- ( ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) /\ y e. ( A " { x } ) ) -> ( F ` <. x , y >. ) = .0. ) |
| 56 |
55
|
mpteq2dva |
|- ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) -> ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) = ( y e. ( A " { x } ) |-> .0. ) ) |
| 57 |
56
|
oveq2d |
|- ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) -> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) = ( W gsum ( y e. ( A " { x } ) |-> .0. ) ) ) |
| 58 |
6
|
cmnmndd |
|- ( ph -> W e. Mnd ) |
| 59 |
8
|
adantr |
|- ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) -> A e. X ) |
| 60 |
59
|
imaexd |
|- ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) -> ( A " { x } ) e. _V ) |
| 61 |
3
|
gsumz |
|- ( ( W e. Mnd /\ ( A " { x } ) e. _V ) -> ( W gsum ( y e. ( A " { x } ) |-> .0. ) ) = .0. ) |
| 62 |
58 60 61
|
syl2an2r |
|- ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) -> ( W gsum ( y e. ( A " { x } ) |-> .0. ) ) = .0. ) |
| 63 |
57 62
|
eqtrd |
|- ( ( ph /\ x e. ( dom A \ dom ( F supp .0. ) ) ) -> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) = .0. ) |
| 64 |
|
dmfi |
|- ( ( F supp .0. ) e. Fin -> dom ( F supp .0. ) e. Fin ) |
| 65 |
30 64
|
syl |
|- ( ph -> dom ( F supp .0. ) e. Fin ) |
| 66 |
6
|
adantr |
|- ( ( ph /\ x e. dom A ) -> W e. CMnd ) |
| 67 |
8
|
adantr |
|- ( ( ph /\ x e. dom A ) -> A e. X ) |
| 68 |
67
|
imaexd |
|- ( ( ph /\ x e. dom A ) -> ( A " { x } ) e. _V ) |
| 69 |
7
|
ad2antrr |
|- ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) -> F : A --> B ) |
| 70 |
22
|
adantl |
|- ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) -> <. x , y >. e. A ) |
| 71 |
69 70
|
ffvelcdmd |
|- ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) -> ( F ` <. x , y >. ) e. B ) |
| 72 |
71
|
fmpttd |
|- ( ( ph /\ x e. dom A ) -> ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) : ( A " { x } ) --> B ) |
| 73 |
68
|
mptexd |
|- ( ( ph /\ x e. dom A ) -> ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) e. _V ) |
| 74 |
72
|
ffnd |
|- ( ( ph /\ x e. dom A ) -> ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) Fn ( A " { x } ) ) |
| 75 |
15
|
a1i |
|- ( ( ph /\ x e. dom A ) -> .0. e. _V ) |
| 76 |
30
|
adantr |
|- ( ( ph /\ x e. dom A ) -> ( F supp .0. ) e. Fin ) |
| 77 |
76 32
|
syl |
|- ( ( ph /\ x e. dom A ) -> ( ( F supp .0. ) " { x } ) e. Fin ) |
| 78 |
|
eqid |
|- ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) = ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) |
| 79 |
|
simp-4l |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> ph ) |
| 80 |
|
simp-4r |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> x e. dom A ) |
| 81 |
|
simpr |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> y = t ) |
| 82 |
|
simpllr |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> t e. ( A " { x } ) ) |
| 83 |
81 82
|
eqeltrd |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> y e. ( A " { x } ) ) |
| 84 |
|
simplr |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> -. t e. ( ( F supp .0. ) " { x } ) ) |
| 85 |
81 84
|
eqneltrd |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> -. y e. ( ( F supp .0. ) " { x } ) ) |
| 86 |
12
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) /\ -. y e. ( ( F supp .0. ) " { x } ) ) -> F Fn A ) |
| 87 |
8
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) /\ -. y e. ( ( F supp .0. ) " { x } ) ) -> A e. X ) |
| 88 |
15
|
a1i |
|- ( ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) /\ -. y e. ( ( F supp .0. ) " { x } ) ) -> .0. e. _V ) |
| 89 |
70
|
adantr |
|- ( ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) /\ -. y e. ( ( F supp .0. ) " { x } ) ) -> <. x , y >. e. A ) |
| 90 |
26
|
con3i |
|- ( -. y e. ( ( F supp .0. ) " { x } ) -> -. <. x , y >. e. ( F supp .0. ) ) |
| 91 |
90
|
adantl |
|- ( ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) /\ -. y e. ( ( F supp .0. ) " { x } ) ) -> -. <. x , y >. e. ( F supp .0. ) ) |
| 92 |
89 91
|
eldifd |
|- ( ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) /\ -. y e. ( ( F supp .0. ) " { x } ) ) -> <. x , y >. e. ( A \ ( F supp .0. ) ) ) |
| 93 |
86 87 88 92
|
fvdifsupp |
|- ( ( ( ( ph /\ x e. dom A ) /\ y e. ( A " { x } ) ) /\ -. y e. ( ( F supp .0. ) " { x } ) ) -> ( F ` <. x , y >. ) = .0. ) |
| 94 |
79 80 83 85 93
|
syl1111anc |
|- ( ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) /\ y = t ) -> ( F ` <. x , y >. ) = .0. ) |
| 95 |
|
simplr |
|- ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) -> t e. ( A " { x } ) ) |
| 96 |
15
|
a1i |
|- ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) -> .0. e. _V ) |
| 97 |
78 94 95 96
|
fvmptd2 |
|- ( ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) /\ -. t e. ( ( F supp .0. ) " { x } ) ) -> ( ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ` t ) = .0. ) |
| 98 |
97
|
ex |
|- ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) -> ( -. t e. ( ( F supp .0. ) " { x } ) -> ( ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ` t ) = .0. ) ) |
| 99 |
98
|
orrd |
|- ( ( ( ph /\ x e. dom A ) /\ t e. ( A " { x } ) ) -> ( t e. ( ( F supp .0. ) " { x } ) \/ ( ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ` t ) = .0. ) ) |
| 100 |
73 74 75 77 99
|
finnzfsuppd |
|- ( ( ph /\ x e. dom A ) -> ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) finSupp .0. ) |
| 101 |
2 3 66 68 72 100
|
gsumcl |
|- ( ( ph /\ x e. dom A ) -> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) e. B ) |
| 102 |
|
dmss |
|- ( ( F supp .0. ) C_ A -> dom ( F supp .0. ) C_ dom A ) |
| 103 |
38 102
|
syl |
|- ( ph -> dom ( F supp .0. ) C_ dom A ) |
| 104 |
2 3 6 45 63 65 101 103
|
gsummptres2 |
|- ( ph -> ( W gsum ( x e. dom A |-> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) = ( W gsum ( x e. dom ( F supp .0. ) |-> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) ) |
| 105 |
7 38
|
feqresmpt |
|- ( ph -> ( F |` ( F supp .0. ) ) = ( z e. ( F supp .0. ) |-> ( F ` z ) ) ) |
| 106 |
105
|
oveq2d |
|- ( ph -> ( W gsum ( F |` ( F supp .0. ) ) ) = ( W gsum ( z e. ( F supp .0. ) |-> ( F ` z ) ) ) ) |
| 107 |
|
ssidd |
|- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) ) |
| 108 |
2 3 6 8 7 107 5
|
gsumres |
|- ( ph -> ( W gsum ( F |` ( F supp .0. ) ) ) = ( W gsum F ) ) |
| 109 |
|
nfcv |
|- F/_ y ( F ` z ) |
| 110 |
|
fveq2 |
|- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) ) |
| 111 |
|
relss |
|- ( ( F supp .0. ) C_ A -> ( Rel A -> Rel ( F supp .0. ) ) ) |
| 112 |
38 4 111
|
sylc |
|- ( ph -> Rel ( F supp .0. ) ) |
| 113 |
7
|
adantr |
|- ( ( ph /\ z e. ( F supp .0. ) ) -> F : A --> B ) |
| 114 |
38
|
sselda |
|- ( ( ph /\ z e. ( F supp .0. ) ) -> z e. A ) |
| 115 |
113 114
|
ffvelcdmd |
|- ( ( ph /\ z e. ( F supp .0. ) ) -> ( F ` z ) e. B ) |
| 116 |
109 1 2 110 112 30 6 115
|
gsummpt2d |
|- ( ph -> ( W gsum ( z e. ( F supp .0. ) |-> ( F ` z ) ) ) = ( W gsum ( x e. dom ( F supp .0. ) |-> ( W gsum ( y e. ( ( F supp .0. ) " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) ) |
| 117 |
106 108 116
|
3eqtr3d |
|- ( ph -> ( W gsum F ) = ( W gsum ( x e. dom ( F supp .0. ) |-> ( W gsum ( y e. ( ( F supp .0. ) " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) ) |
| 118 |
44 104 117
|
3eqtr4rd |
|- ( ph -> ( W gsum F ) = ( W gsum ( x e. dom A |-> ( W gsum ( y e. ( A " { x } ) |-> ( F ` <. x , y >. ) ) ) ) ) ) |