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 >. ) ) ) ) ) ) |