Step |
Hyp |
Ref |
Expression |
1 |
|
gsumhashmul.b |
|- B = ( Base ` G ) |
2 |
|
gsumhashmul.z |
|- .0. = ( 0g ` G ) |
3 |
|
gsumhashmul.x |
|- .x. = ( .g ` G ) |
4 |
|
gsumhashmul.g |
|- ( ph -> G e. CMnd ) |
5 |
|
gsumhashmul.f |
|- ( ph -> F : A --> B ) |
6 |
|
gsumhashmul.1 |
|- ( ph -> F finSupp .0. ) |
7 |
|
suppssdm |
|- ( F supp .0. ) C_ dom F |
8 |
7 5
|
fssdm |
|- ( ph -> ( F supp .0. ) C_ A ) |
9 |
5 8
|
feqresmpt |
|- ( ph -> ( F |` ( F supp .0. ) ) = ( x e. ( F supp .0. ) |-> ( F ` x ) ) ) |
10 |
9
|
oveq2d |
|- ( ph -> ( G gsum ( F |` ( F supp .0. ) ) ) = ( G gsum ( x e. ( F supp .0. ) |-> ( F ` x ) ) ) ) |
11 |
|
relfsupp |
|- Rel finSupp |
12 |
11
|
brrelex1i |
|- ( F finSupp .0. -> F e. _V ) |
13 |
6 12
|
syl |
|- ( ph -> F e. _V ) |
14 |
5
|
ffnd |
|- ( ph -> F Fn A ) |
15 |
13 14
|
fndmexd |
|- ( ph -> A e. _V ) |
16 |
|
ssidd |
|- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) ) |
17 |
1 2 4 15 5 16 6
|
gsumres |
|- ( ph -> ( G gsum ( F |` ( F supp .0. ) ) ) = ( G gsum F ) ) |
18 |
|
nfcv |
|- F/_ x ( F ` ( 1st ` z ) ) |
19 |
|
fveq2 |
|- ( x = ( 1st ` z ) -> ( F ` x ) = ( F ` ( 1st ` z ) ) ) |
20 |
6
|
fsuppimpd |
|- ( ph -> ( F supp .0. ) e. Fin ) |
21 |
|
ssidd |
|- ( ph -> B C_ B ) |
22 |
5
|
adantr |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> F : A --> B ) |
23 |
8
|
sselda |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> x e. A ) |
24 |
22 23
|
ffvelrnd |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> ( F ` x ) e. B ) |
25 |
5
|
ffund |
|- ( ph -> Fun F ) |
26 |
|
funrel |
|- ( Fun F -> Rel F ) |
27 |
|
reldif |
|- ( Rel F -> Rel ( F \ ( _V X. { .0. } ) ) ) |
28 |
25 26 27
|
3syl |
|- ( ph -> Rel ( F \ ( _V X. { .0. } ) ) ) |
29 |
|
1stdm |
|- ( ( Rel ( F \ ( _V X. { .0. } ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> ( 1st ` z ) e. dom ( F \ ( _V X. { .0. } ) ) ) |
30 |
28 29
|
sylan |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> ( 1st ` z ) e. dom ( F \ ( _V X. { .0. } ) ) ) |
31 |
2
|
fvexi |
|- .0. e. _V |
32 |
31
|
a1i |
|- ( ph -> .0. e. _V ) |
33 |
|
fressupp |
|- ( ( Fun F /\ F e. _V /\ .0. e. _V ) -> ( F |` ( F supp .0. ) ) = ( F \ ( _V X. { .0. } ) ) ) |
34 |
25 13 32 33
|
syl3anc |
|- ( ph -> ( F |` ( F supp .0. ) ) = ( F \ ( _V X. { .0. } ) ) ) |
35 |
34
|
dmeqd |
|- ( ph -> dom ( F |` ( F supp .0. ) ) = dom ( F \ ( _V X. { .0. } ) ) ) |
36 |
7
|
a1i |
|- ( ph -> ( F supp .0. ) C_ dom F ) |
37 |
|
ssdmres |
|- ( ( F supp .0. ) C_ dom F <-> dom ( F |` ( F supp .0. ) ) = ( F supp .0. ) ) |
38 |
36 37
|
sylib |
|- ( ph -> dom ( F |` ( F supp .0. ) ) = ( F supp .0. ) ) |
39 |
35 38
|
eqtr3d |
|- ( ph -> dom ( F \ ( _V X. { .0. } ) ) = ( F supp .0. ) ) |
40 |
39
|
adantr |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> dom ( F \ ( _V X. { .0. } ) ) = ( F supp .0. ) ) |
41 |
30 40
|
eleqtrd |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> ( 1st ` z ) e. ( F supp .0. ) ) |
42 |
25
|
funresd |
|- ( ph -> Fun ( F |` ( F supp .0. ) ) ) |
43 |
42
|
adantr |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> Fun ( F |` ( F supp .0. ) ) ) |
44 |
38
|
eleq2d |
|- ( ph -> ( x e. dom ( F |` ( F supp .0. ) ) <-> x e. ( F supp .0. ) ) ) |
45 |
44
|
biimpar |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> x e. dom ( F |` ( F supp .0. ) ) ) |
46 |
|
simpr |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> x e. ( F supp .0. ) ) |
47 |
46
|
fvresd |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> ( ( F |` ( F supp .0. ) ) ` x ) = ( F ` x ) ) |
48 |
|
funopfvb |
|- ( ( Fun ( F |` ( F supp .0. ) ) /\ x e. dom ( F |` ( F supp .0. ) ) ) -> ( ( ( F |` ( F supp .0. ) ) ` x ) = ( F ` x ) <-> <. x , ( F ` x ) >. e. ( F |` ( F supp .0. ) ) ) ) |
49 |
48
|
biimpa |
|- ( ( ( Fun ( F |` ( F supp .0. ) ) /\ x e. dom ( F |` ( F supp .0. ) ) ) /\ ( ( F |` ( F supp .0. ) ) ` x ) = ( F ` x ) ) -> <. x , ( F ` x ) >. e. ( F |` ( F supp .0. ) ) ) |
50 |
43 45 47 49
|
syl21anc |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> <. x , ( F ` x ) >. e. ( F |` ( F supp .0. ) ) ) |
51 |
34
|
adantr |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> ( F |` ( F supp .0. ) ) = ( F \ ( _V X. { .0. } ) ) ) |
52 |
50 51
|
eleqtrd |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> <. x , ( F ` x ) >. e. ( F \ ( _V X. { .0. } ) ) ) |
53 |
|
eqeq2 |
|- ( v = <. x , ( F ` x ) >. -> ( z = v <-> z = <. x , ( F ` x ) >. ) ) |
54 |
53
|
bibi2d |
|- ( v = <. x , ( F ` x ) >. -> ( ( x = ( 1st ` z ) <-> z = v ) <-> ( x = ( 1st ` z ) <-> z = <. x , ( F ` x ) >. ) ) ) |
55 |
54
|
ralbidv |
|- ( v = <. x , ( F ` x ) >. -> ( A. z e. ( F \ ( _V X. { .0. } ) ) ( x = ( 1st ` z ) <-> z = v ) <-> A. z e. ( F \ ( _V X. { .0. } ) ) ( x = ( 1st ` z ) <-> z = <. x , ( F ` x ) >. ) ) ) |
56 |
55
|
adantl |
|- ( ( ( ph /\ x e. ( F supp .0. ) ) /\ v = <. x , ( F ` x ) >. ) -> ( A. z e. ( F \ ( _V X. { .0. } ) ) ( x = ( 1st ` z ) <-> z = v ) <-> A. z e. ( F \ ( _V X. { .0. } ) ) ( x = ( 1st ` z ) <-> z = <. x , ( F ` x ) >. ) ) ) |
57 |
|
fvexd |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> ( 2nd ` z ) e. _V ) |
58 |
28
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> Rel ( F \ ( _V X. { .0. } ) ) ) |
59 |
|
simplr |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> z e. ( F \ ( _V X. { .0. } ) ) ) |
60 |
|
1st2nd |
|- ( ( Rel ( F \ ( _V X. { .0. } ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. ) |
61 |
58 59 60
|
syl2anc |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. ) |
62 |
|
opeq1 |
|- ( x = ( 1st ` z ) -> <. x , ( 2nd ` z ) >. = <. ( 1st ` z ) , ( 2nd ` z ) >. ) |
63 |
62
|
adantl |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> <. x , ( 2nd ` z ) >. = <. ( 1st ` z ) , ( 2nd ` z ) >. ) |
64 |
61 63
|
eqtr4d |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> z = <. x , ( 2nd ` z ) >. ) |
65 |
|
difssd |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> ( F \ ( _V X. { .0. } ) ) C_ F ) |
66 |
65
|
sselda |
|- ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> z e. F ) |
67 |
66
|
adantr |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> z e. F ) |
68 |
64 67
|
eqeltrrd |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> <. x , ( 2nd ` z ) >. e. F ) |
69 |
64 68
|
jca |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> ( z = <. x , ( 2nd ` z ) >. /\ <. x , ( 2nd ` z ) >. e. F ) ) |
70 |
|
opeq2 |
|- ( y = ( 2nd ` z ) -> <. x , y >. = <. x , ( 2nd ` z ) >. ) |
71 |
70
|
eqeq2d |
|- ( y = ( 2nd ` z ) -> ( z = <. x , y >. <-> z = <. x , ( 2nd ` z ) >. ) ) |
72 |
70
|
eleq1d |
|- ( y = ( 2nd ` z ) -> ( <. x , y >. e. F <-> <. x , ( 2nd ` z ) >. e. F ) ) |
73 |
71 72
|
anbi12d |
|- ( y = ( 2nd ` z ) -> ( ( z = <. x , y >. /\ <. x , y >. e. F ) <-> ( z = <. x , ( 2nd ` z ) >. /\ <. x , ( 2nd ` z ) >. e. F ) ) ) |
74 |
57 69 73
|
spcedv |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> E. y ( z = <. x , y >. /\ <. x , y >. e. F ) ) |
75 |
|
vex |
|- x e. _V |
76 |
75
|
elsnres |
|- ( z e. ( F |` { x } ) <-> E. y ( z = <. x , y >. /\ <. x , y >. e. F ) ) |
77 |
74 76
|
sylibr |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> z e. ( F |` { x } ) ) |
78 |
14
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> F Fn A ) |
79 |
23
|
ad2antrr |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> x e. A ) |
80 |
|
fnressn |
|- ( ( F Fn A /\ x e. A ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) |
81 |
78 79 80
|
syl2anc |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> ( F |` { x } ) = { <. x , ( F ` x ) >. } ) |
82 |
77 81
|
eleqtrd |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> z e. { <. x , ( F ` x ) >. } ) |
83 |
|
elsni |
|- ( z e. { <. x , ( F ` x ) >. } -> z = <. x , ( F ` x ) >. ) |
84 |
82 83
|
syl |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ x = ( 1st ` z ) ) -> z = <. x , ( F ` x ) >. ) |
85 |
|
simpr |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ z = <. x , ( F ` x ) >. ) -> z = <. x , ( F ` x ) >. ) |
86 |
85
|
fveq2d |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ z = <. x , ( F ` x ) >. ) -> ( 1st ` z ) = ( 1st ` <. x , ( F ` x ) >. ) ) |
87 |
|
fvex |
|- ( F ` x ) e. _V |
88 |
75 87
|
op1st |
|- ( 1st ` <. x , ( F ` x ) >. ) = x |
89 |
86 88
|
eqtr2di |
|- ( ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ z = <. x , ( F ` x ) >. ) -> x = ( 1st ` z ) ) |
90 |
84 89
|
impbida |
|- ( ( ( ph /\ x e. ( F supp .0. ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> ( x = ( 1st ` z ) <-> z = <. x , ( F ` x ) >. ) ) |
91 |
90
|
ralrimiva |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> A. z e. ( F \ ( _V X. { .0. } ) ) ( x = ( 1st ` z ) <-> z = <. x , ( F ` x ) >. ) ) |
92 |
52 56 91
|
rspcedvd |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> E. v e. ( F \ ( _V X. { .0. } ) ) A. z e. ( F \ ( _V X. { .0. } ) ) ( x = ( 1st ` z ) <-> z = v ) ) |
93 |
|
reu6 |
|- ( E! z e. ( F \ ( _V X. { .0. } ) ) x = ( 1st ` z ) <-> E. v e. ( F \ ( _V X. { .0. } ) ) A. z e. ( F \ ( _V X. { .0. } ) ) ( x = ( 1st ` z ) <-> z = v ) ) |
94 |
92 93
|
sylibr |
|- ( ( ph /\ x e. ( F supp .0. ) ) -> E! z e. ( F \ ( _V X. { .0. } ) ) x = ( 1st ` z ) ) |
95 |
18 1 2 19 4 20 21 24 41 94
|
gsummptf1o |
|- ( ph -> ( G gsum ( x e. ( F supp .0. ) |-> ( F ` x ) ) ) = ( G gsum ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( F ` ( 1st ` z ) ) ) ) ) |
96 |
10 17 95
|
3eqtr3d |
|- ( ph -> ( G gsum F ) = ( G gsum ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( F ` ( 1st ` z ) ) ) ) ) |
97 |
|
simpr |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> z e. ( F \ ( _V X. { .0. } ) ) ) |
98 |
97
|
eldifad |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> z e. F ) |
99 |
|
funfv1st2nd |
|- ( ( Fun F /\ z e. F ) -> ( F ` ( 1st ` z ) ) = ( 2nd ` z ) ) |
100 |
25 98 99
|
syl2an2r |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> ( F ` ( 1st ` z ) ) = ( 2nd ` z ) ) |
101 |
100
|
mpteq2dva |
|- ( ph -> ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( F ` ( 1st ` z ) ) ) = ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( 2nd ` z ) ) ) |
102 |
101
|
oveq2d |
|- ( ph -> ( G gsum ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( F ` ( 1st ` z ) ) ) ) = ( G gsum ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( 2nd ` z ) ) ) ) |
103 |
96 102
|
eqtrd |
|- ( ph -> ( G gsum F ) = ( G gsum ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( 2nd ` z ) ) ) ) |
104 |
|
nfcv |
|- F/_ z ( 1st ` t ) |
105 |
|
fvex |
|- ( 2nd ` t ) e. _V |
106 |
|
fvex |
|- ( 1st ` t ) e. _V |
107 |
105 106
|
op2ndd |
|- ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. -> ( 2nd ` z ) = ( 1st ` t ) ) |
108 |
|
resfnfinfin |
|- ( ( F Fn A /\ ( F supp .0. ) e. Fin ) -> ( F |` ( F supp .0. ) ) e. Fin ) |
109 |
14 20 108
|
syl2anc |
|- ( ph -> ( F |` ( F supp .0. ) ) e. Fin ) |
110 |
34 109
|
eqeltrrd |
|- ( ph -> ( F \ ( _V X. { .0. } ) ) e. Fin ) |
111 |
34
|
rneqd |
|- ( ph -> ran ( F |` ( F supp .0. ) ) = ran ( F \ ( _V X. { .0. } ) ) ) |
112 |
|
rnresss |
|- ran ( F |` ( F supp .0. ) ) C_ ran F |
113 |
5
|
frnd |
|- ( ph -> ran F C_ B ) |
114 |
112 113
|
sstrid |
|- ( ph -> ran ( F |` ( F supp .0. ) ) C_ B ) |
115 |
111 114
|
eqsstrrd |
|- ( ph -> ran ( F \ ( _V X. { .0. } ) ) C_ B ) |
116 |
|
2ndrn |
|- ( ( Rel ( F \ ( _V X. { .0. } ) ) /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> ( 2nd ` z ) e. ran ( F \ ( _V X. { .0. } ) ) ) |
117 |
28 116
|
sylan |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> ( 2nd ` z ) e. ran ( F \ ( _V X. { .0. } ) ) ) |
118 |
|
relcnv |
|- Rel `' F |
119 |
|
reldif |
|- ( Rel `' F -> Rel ( `' F \ ( { .0. } X. _V ) ) ) |
120 |
118 119
|
mp1i |
|- ( ph -> Rel ( `' F \ ( { .0. } X. _V ) ) ) |
121 |
|
1st2nd |
|- ( ( Rel ( `' F \ ( { .0. } X. _V ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) -> t = <. ( 1st ` t ) , ( 2nd ` t ) >. ) |
122 |
120 121
|
sylan |
|- ( ( ph /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) -> t = <. ( 1st ` t ) , ( 2nd ` t ) >. ) |
123 |
|
cnvdif |
|- `' ( F \ ( _V X. { .0. } ) ) = ( `' F \ `' ( _V X. { .0. } ) ) |
124 |
|
cnvxp |
|- `' ( _V X. { .0. } ) = ( { .0. } X. _V ) |
125 |
124
|
difeq2i |
|- ( `' F \ `' ( _V X. { .0. } ) ) = ( `' F \ ( { .0. } X. _V ) ) |
126 |
123 125
|
eqtri |
|- `' ( F \ ( _V X. { .0. } ) ) = ( `' F \ ( { .0. } X. _V ) ) |
127 |
126
|
eqimss2i |
|- ( `' F \ ( { .0. } X. _V ) ) C_ `' ( F \ ( _V X. { .0. } ) ) |
128 |
127
|
a1i |
|- ( ph -> ( `' F \ ( { .0. } X. _V ) ) C_ `' ( F \ ( _V X. { .0. } ) ) ) |
129 |
128
|
sselda |
|- ( ( ph /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) -> t e. `' ( F \ ( _V X. { .0. } ) ) ) |
130 |
122 129
|
eqeltrrd |
|- ( ( ph /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. `' ( F \ ( _V X. { .0. } ) ) ) |
131 |
106 105
|
opelcnv |
|- ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. `' ( F \ ( _V X. { .0. } ) ) <-> <. ( 2nd ` t ) , ( 1st ` t ) >. e. ( F \ ( _V X. { .0. } ) ) ) |
132 |
130 131
|
sylib |
|- ( ( ph /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) -> <. ( 2nd ` t ) , ( 1st ` t ) >. e. ( F \ ( _V X. { .0. } ) ) ) |
133 |
28
|
adantr |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> Rel ( F \ ( _V X. { .0. } ) ) ) |
134 |
|
eqidd |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> U. `' { z } = U. `' { z } ) |
135 |
|
cnvf1olem |
|- ( ( Rel ( F \ ( _V X. { .0. } ) ) /\ ( z e. ( F \ ( _V X. { .0. } ) ) /\ U. `' { z } = U. `' { z } ) ) -> ( U. `' { z } e. `' ( F \ ( _V X. { .0. } ) ) /\ z = U. `' { U. `' { z } } ) ) |
136 |
135
|
simpld |
|- ( ( Rel ( F \ ( _V X. { .0. } ) ) /\ ( z e. ( F \ ( _V X. { .0. } ) ) /\ U. `' { z } = U. `' { z } ) ) -> U. `' { z } e. `' ( F \ ( _V X. { .0. } ) ) ) |
137 |
133 97 134 136
|
syl12anc |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> U. `' { z } e. `' ( F \ ( _V X. { .0. } ) ) ) |
138 |
137 126
|
eleqtrdi |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> U. `' { z } e. ( `' F \ ( { .0. } X. _V ) ) ) |
139 |
|
eqeq2 |
|- ( u = U. `' { z } -> ( t = u <-> t = U. `' { z } ) ) |
140 |
139
|
bibi2d |
|- ( u = U. `' { z } -> ( ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = u ) <-> ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = U. `' { z } ) ) ) |
141 |
140
|
ralbidv |
|- ( u = U. `' { z } -> ( A. t e. ( `' F \ ( { .0. } X. _V ) ) ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = u ) <-> A. t e. ( `' F \ ( { .0. } X. _V ) ) ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = U. `' { z } ) ) ) |
142 |
141
|
adantl |
|- ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ u = U. `' { z } ) -> ( A. t e. ( `' F \ ( { .0. } X. _V ) ) ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = u ) <-> A. t e. ( `' F \ ( { .0. } X. _V ) ) ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = U. `' { z } ) ) ) |
143 |
118 119
|
mp1i |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> Rel ( `' F \ ( { .0. } X. _V ) ) ) |
144 |
|
simplr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> t e. ( `' F \ ( { .0. } X. _V ) ) ) |
145 |
|
simpr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) |
146 |
|
df-rel |
|- ( Rel ( `' F \ ( { .0. } X. _V ) ) <-> ( `' F \ ( { .0. } X. _V ) ) C_ ( _V X. _V ) ) |
147 |
120 146
|
sylib |
|- ( ph -> ( `' F \ ( { .0. } X. _V ) ) C_ ( _V X. _V ) ) |
148 |
147
|
ad3antrrr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> ( `' F \ ( { .0. } X. _V ) ) C_ ( _V X. _V ) ) |
149 |
148 144
|
sseldd |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> t e. ( _V X. _V ) ) |
150 |
|
2nd1st |
|- ( t e. ( _V X. _V ) -> U. `' { t } = <. ( 2nd ` t ) , ( 1st ` t ) >. ) |
151 |
149 150
|
syl |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> U. `' { t } = <. ( 2nd ` t ) , ( 1st ` t ) >. ) |
152 |
145 151
|
eqtr4d |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> z = U. `' { t } ) |
153 |
|
cnvf1olem |
|- ( ( Rel ( `' F \ ( { .0. } X. _V ) ) /\ ( t e. ( `' F \ ( { .0. } X. _V ) ) /\ z = U. `' { t } ) ) -> ( z e. `' ( `' F \ ( { .0. } X. _V ) ) /\ t = U. `' { z } ) ) |
154 |
153
|
simprd |
|- ( ( Rel ( `' F \ ( { .0. } X. _V ) ) /\ ( t e. ( `' F \ ( { .0. } X. _V ) ) /\ z = U. `' { t } ) ) -> t = U. `' { z } ) |
155 |
143 144 152 154
|
syl12anc |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) -> t = U. `' { z } ) |
156 |
28
|
ad3antrrr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> Rel ( F \ ( _V X. { .0. } ) ) ) |
157 |
97
|
ad2antrr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> z e. ( F \ ( _V X. { .0. } ) ) ) |
158 |
|
simpr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> t = U. `' { z } ) |
159 |
|
cnvf1olem |
|- ( ( Rel ( F \ ( _V X. { .0. } ) ) /\ ( z e. ( F \ ( _V X. { .0. } ) ) /\ t = U. `' { z } ) ) -> ( t e. `' ( F \ ( _V X. { .0. } ) ) /\ z = U. `' { t } ) ) |
160 |
159
|
simprd |
|- ( ( Rel ( F \ ( _V X. { .0. } ) ) /\ ( z e. ( F \ ( _V X. { .0. } ) ) /\ t = U. `' { z } ) ) -> z = U. `' { t } ) |
161 |
156 157 158 160
|
syl12anc |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> z = U. `' { t } ) |
162 |
147
|
ad3antrrr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> ( `' F \ ( { .0. } X. _V ) ) C_ ( _V X. _V ) ) |
163 |
|
simplr |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> t e. ( `' F \ ( { .0. } X. _V ) ) ) |
164 |
162 163
|
sseldd |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> t e. ( _V X. _V ) ) |
165 |
164 150
|
syl |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> U. `' { t } = <. ( 2nd ` t ) , ( 1st ` t ) >. ) |
166 |
161 165
|
eqtrd |
|- ( ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) /\ t = U. `' { z } ) -> z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) |
167 |
155 166
|
impbida |
|- ( ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) /\ t e. ( `' F \ ( { .0. } X. _V ) ) ) -> ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = U. `' { z } ) ) |
168 |
167
|
ralrimiva |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> A. t e. ( `' F \ ( { .0. } X. _V ) ) ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = U. `' { z } ) ) |
169 |
138 142 168
|
rspcedvd |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> E. u e. ( `' F \ ( { .0. } X. _V ) ) A. t e. ( `' F \ ( { .0. } X. _V ) ) ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = u ) ) |
170 |
|
reu6 |
|- ( E! t e. ( `' F \ ( { .0. } X. _V ) ) z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> E. u e. ( `' F \ ( { .0. } X. _V ) ) A. t e. ( `' F \ ( { .0. } X. _V ) ) ( z = <. ( 2nd ` t ) , ( 1st ` t ) >. <-> t = u ) ) |
171 |
169 170
|
sylibr |
|- ( ( ph /\ z e. ( F \ ( _V X. { .0. } ) ) ) -> E! t e. ( `' F \ ( { .0. } X. _V ) ) z = <. ( 2nd ` t ) , ( 1st ` t ) >. ) |
172 |
104 1 2 107 4 110 115 117 132 171
|
gsummptf1o |
|- ( ph -> ( G gsum ( z e. ( F \ ( _V X. { .0. } ) ) |-> ( 2nd ` z ) ) ) = ( G gsum ( t e. ( `' F \ ( { .0. } X. _V ) ) |-> ( 1st ` t ) ) ) ) |
173 |
|
fveq2 |
|- ( t = z -> ( 1st ` t ) = ( 1st ` z ) ) |
174 |
173
|
cbvmptv |
|- ( t e. ( `' F \ ( { .0. } X. _V ) ) |-> ( 1st ` t ) ) = ( z e. ( `' F \ ( { .0. } X. _V ) ) |-> ( 1st ` z ) ) |
175 |
34
|
cnveqd |
|- ( ph -> `' ( F |` ( F supp .0. ) ) = `' ( F \ ( _V X. { .0. } ) ) ) |
176 |
175 126
|
eqtr2di |
|- ( ph -> ( `' F \ ( { .0. } X. _V ) ) = `' ( F |` ( F supp .0. ) ) ) |
177 |
176
|
mpteq1d |
|- ( ph -> ( z e. ( `' F \ ( { .0. } X. _V ) ) |-> ( 1st ` z ) ) = ( z e. `' ( F |` ( F supp .0. ) ) |-> ( 1st ` z ) ) ) |
178 |
174 177
|
syl5eq |
|- ( ph -> ( t e. ( `' F \ ( { .0. } X. _V ) ) |-> ( 1st ` t ) ) = ( z e. `' ( F |` ( F supp .0. ) ) |-> ( 1st ` z ) ) ) |
179 |
178
|
oveq2d |
|- ( ph -> ( G gsum ( t e. ( `' F \ ( { .0. } X. _V ) ) |-> ( 1st ` t ) ) ) = ( G gsum ( z e. `' ( F |` ( F supp .0. ) ) |-> ( 1st ` z ) ) ) ) |
180 |
103 172 179
|
3eqtrd |
|- ( ph -> ( G gsum F ) = ( G gsum ( z e. `' ( F |` ( F supp .0. ) ) |-> ( 1st ` z ) ) ) ) |
181 |
|
nfcv |
|- F/_ y ( 1st ` z ) |
182 |
|
nfv |
|- F/ x ph |
183 |
|
vex |
|- y e. _V |
184 |
75 183
|
op1std |
|- ( z = <. x , y >. -> ( 1st ` z ) = x ) |
185 |
|
relcnv |
|- Rel `' ( F |` ( F supp .0. ) ) |
186 |
185
|
a1i |
|- ( ph -> Rel `' ( F |` ( F supp .0. ) ) ) |
187 |
|
cnvfi |
|- ( ( F |` ( F supp .0. ) ) e. Fin -> `' ( F |` ( F supp .0. ) ) e. Fin ) |
188 |
109 187
|
syl |
|- ( ph -> `' ( F |` ( F supp .0. ) ) e. Fin ) |
189 |
113
|
adantr |
|- ( ( ph /\ z e. `' ( F |` ( F supp .0. ) ) ) -> ran F C_ B ) |
190 |
185
|
a1i |
|- ( ( ph /\ z e. `' ( F |` ( F supp .0. ) ) ) -> Rel `' ( F |` ( F supp .0. ) ) ) |
191 |
|
simpr |
|- ( ( ph /\ z e. `' ( F |` ( F supp .0. ) ) ) -> z e. `' ( F |` ( F supp .0. ) ) ) |
192 |
|
1stdm |
|- ( ( Rel `' ( F |` ( F supp .0. ) ) /\ z e. `' ( F |` ( F supp .0. ) ) ) -> ( 1st ` z ) e. dom `' ( F |` ( F supp .0. ) ) ) |
193 |
190 191 192
|
syl2anc |
|- ( ( ph /\ z e. `' ( F |` ( F supp .0. ) ) ) -> ( 1st ` z ) e. dom `' ( F |` ( F supp .0. ) ) ) |
194 |
|
df-rn |
|- ran ( F |` ( F supp .0. ) ) = dom `' ( F |` ( F supp .0. ) ) |
195 |
193 194
|
eleqtrrdi |
|- ( ( ph /\ z e. `' ( F |` ( F supp .0. ) ) ) -> ( 1st ` z ) e. ran ( F |` ( F supp .0. ) ) ) |
196 |
112 195
|
sselid |
|- ( ( ph /\ z e. `' ( F |` ( F supp .0. ) ) ) -> ( 1st ` z ) e. ran F ) |
197 |
189 196
|
sseldd |
|- ( ( ph /\ z e. `' ( F |` ( F supp .0. ) ) ) -> ( 1st ` z ) e. B ) |
198 |
181 182 1 184 186 188 4 197
|
gsummpt2d |
|- ( ph -> ( G gsum ( z e. `' ( F |` ( F supp .0. ) ) |-> ( 1st ` z ) ) ) = ( G gsum ( x e. dom `' ( F |` ( F supp .0. ) ) |-> ( G gsum ( y e. ( `' ( F |` ( F supp .0. ) ) " { x } ) |-> x ) ) ) ) ) |
199 |
|
df-ima |
|- ( F " ( F supp .0. ) ) = ran ( F |` ( F supp .0. ) ) |
200 |
|
supppreima |
|- ( ( Fun F /\ F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( ran F \ { .0. } ) ) ) |
201 |
25 13 32 200
|
syl3anc |
|- ( ph -> ( F supp .0. ) = ( `' F " ( ran F \ { .0. } ) ) ) |
202 |
201
|
imaeq2d |
|- ( ph -> ( F " ( F supp .0. ) ) = ( F " ( `' F " ( ran F \ { .0. } ) ) ) ) |
203 |
199 202
|
eqtr3id |
|- ( ph -> ran ( F |` ( F supp .0. ) ) = ( F " ( `' F " ( ran F \ { .0. } ) ) ) ) |
204 |
|
funimacnv |
|- ( Fun F -> ( F " ( `' F " ( ran F \ { .0. } ) ) ) = ( ( ran F \ { .0. } ) i^i ran F ) ) |
205 |
25 204
|
syl |
|- ( ph -> ( F " ( `' F " ( ran F \ { .0. } ) ) ) = ( ( ran F \ { .0. } ) i^i ran F ) ) |
206 |
|
difssd |
|- ( ph -> ( ran F \ { .0. } ) C_ ran F ) |
207 |
|
df-ss |
|- ( ( ran F \ { .0. } ) C_ ran F <-> ( ( ran F \ { .0. } ) i^i ran F ) = ( ran F \ { .0. } ) ) |
208 |
206 207
|
sylib |
|- ( ph -> ( ( ran F \ { .0. } ) i^i ran F ) = ( ran F \ { .0. } ) ) |
209 |
203 205 208
|
3eqtrd |
|- ( ph -> ran ( F |` ( F supp .0. ) ) = ( ran F \ { .0. } ) ) |
210 |
194 209
|
eqtr3id |
|- ( ph -> dom `' ( F |` ( F supp .0. ) ) = ( ran F \ { .0. } ) ) |
211 |
4
|
cmnmndd |
|- ( ph -> G e. Mnd ) |
212 |
211
|
adantr |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> G e. Mnd ) |
213 |
109
|
adantr |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( F |` ( F supp .0. ) ) e. Fin ) |
214 |
|
imafi2 |
|- ( `' ( F |` ( F supp .0. ) ) e. Fin -> ( `' ( F |` ( F supp .0. ) ) " { x } ) e. Fin ) |
215 |
213 187 214
|
3syl |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( `' ( F |` ( F supp .0. ) ) " { x } ) e. Fin ) |
216 |
194 114
|
eqsstrrid |
|- ( ph -> dom `' ( F |` ( F supp .0. ) ) C_ B ) |
217 |
216
|
sselda |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> x e. B ) |
218 |
1 3
|
gsumconst |
|- ( ( G e. Mnd /\ ( `' ( F |` ( F supp .0. ) ) " { x } ) e. Fin /\ x e. B ) -> ( G gsum ( y e. ( `' ( F |` ( F supp .0. ) ) " { x } ) |-> x ) ) = ( ( # ` ( `' ( F |` ( F supp .0. ) ) " { x } ) ) .x. x ) ) |
219 |
212 215 217 218
|
syl3anc |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( G gsum ( y e. ( `' ( F |` ( F supp .0. ) ) " { x } ) |-> x ) ) = ( ( # ` ( `' ( F |` ( F supp .0. ) ) " { x } ) ) .x. x ) ) |
220 |
|
cnvresima |
|- ( `' ( F |` ( F supp .0. ) ) " { x } ) = ( ( `' F " { x } ) i^i ( F supp .0. ) ) |
221 |
210
|
eleq2d |
|- ( ph -> ( x e. dom `' ( F |` ( F supp .0. ) ) <-> x e. ( ran F \ { .0. } ) ) ) |
222 |
221
|
biimpa |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> x e. ( ran F \ { .0. } ) ) |
223 |
222
|
snssd |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> { x } C_ ( ran F \ { .0. } ) ) |
224 |
|
sspreima |
|- ( ( Fun F /\ { x } C_ ( ran F \ { .0. } ) ) -> ( `' F " { x } ) C_ ( `' F " ( ran F \ { .0. } ) ) ) |
225 |
25 223 224
|
syl2an2r |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( `' F " { x } ) C_ ( `' F " ( ran F \ { .0. } ) ) ) |
226 |
201
|
adantr |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( F supp .0. ) = ( `' F " ( ran F \ { .0. } ) ) ) |
227 |
225 226
|
sseqtrrd |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( `' F " { x } ) C_ ( F supp .0. ) ) |
228 |
|
df-ss |
|- ( ( `' F " { x } ) C_ ( F supp .0. ) <-> ( ( `' F " { x } ) i^i ( F supp .0. ) ) = ( `' F " { x } ) ) |
229 |
227 228
|
sylib |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( ( `' F " { x } ) i^i ( F supp .0. ) ) = ( `' F " { x } ) ) |
230 |
220 229
|
eqtr2id |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( `' F " { x } ) = ( `' ( F |` ( F supp .0. ) ) " { x } ) ) |
231 |
230
|
fveq2d |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( # ` ( `' F " { x } ) ) = ( # ` ( `' ( F |` ( F supp .0. ) ) " { x } ) ) ) |
232 |
231
|
oveq1d |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( ( # ` ( `' F " { x } ) ) .x. x ) = ( ( # ` ( `' ( F |` ( F supp .0. ) ) " { x } ) ) .x. x ) ) |
233 |
219 232
|
eqtr4d |
|- ( ( ph /\ x e. dom `' ( F |` ( F supp .0. ) ) ) -> ( G gsum ( y e. ( `' ( F |` ( F supp .0. ) ) " { x } ) |-> x ) ) = ( ( # ` ( `' F " { x } ) ) .x. x ) ) |
234 |
210 233
|
mpteq12dva |
|- ( ph -> ( x e. dom `' ( F |` ( F supp .0. ) ) |-> ( G gsum ( y e. ( `' ( F |` ( F supp .0. ) ) " { x } ) |-> x ) ) ) = ( x e. ( ran F \ { .0. } ) |-> ( ( # ` ( `' F " { x } ) ) .x. x ) ) ) |
235 |
234
|
oveq2d |
|- ( ph -> ( G gsum ( x e. dom `' ( F |` ( F supp .0. ) ) |-> ( G gsum ( y e. ( `' ( F |` ( F supp .0. ) ) " { x } ) |-> x ) ) ) ) = ( G gsum ( x e. ( ran F \ { .0. } ) |-> ( ( # ` ( `' F " { x } ) ) .x. x ) ) ) ) |
236 |
180 198 235
|
3eqtrd |
|- ( ph -> ( G gsum F ) = ( G gsum ( x e. ( ran F \ { .0. } ) |-> ( ( # ` ( `' F " { x } ) ) .x. x ) ) ) ) |