Step |
Hyp |
Ref |
Expression |
1 |
|
fsumcn.3 |
|- K = ( TopOpen ` CCfld ) |
2 |
|
fsumcn.4 |
|- ( ph -> J e. ( TopOn ` X ) ) |
3 |
|
fsumcn.5 |
|- ( ph -> A e. Fin ) |
4 |
|
fsum2cn.7 |
|- ( ph -> L e. ( TopOn ` Y ) ) |
5 |
|
fsum2cn.8 |
|- ( ( ph /\ k e. A ) -> ( x e. X , y e. Y |-> B ) e. ( ( J tX L ) Cn K ) ) |
6 |
|
nfcv |
|- F/_ u sum_ k e. A B |
7 |
|
nfcv |
|- F/_ v sum_ k e. A B |
8 |
|
nfcv |
|- F/_ x A |
9 |
|
nfcv |
|- F/_ x v |
10 |
|
nfcsb1v |
|- F/_ x [_ u / x ]_ B |
11 |
9 10
|
nfcsbw |
|- F/_ x [_ v / y ]_ [_ u / x ]_ B |
12 |
8 11
|
nfsum |
|- F/_ x sum_ k e. A [_ v / y ]_ [_ u / x ]_ B |
13 |
|
nfcv |
|- F/_ y A |
14 |
|
nfcsb1v |
|- F/_ y [_ v / y ]_ [_ u / x ]_ B |
15 |
13 14
|
nfsum |
|- F/_ y sum_ k e. A [_ v / y ]_ [_ u / x ]_ B |
16 |
|
csbeq1a |
|- ( x = u -> B = [_ u / x ]_ B ) |
17 |
|
csbeq1a |
|- ( y = v -> [_ u / x ]_ B = [_ v / y ]_ [_ u / x ]_ B ) |
18 |
16 17
|
sylan9eq |
|- ( ( x = u /\ y = v ) -> B = [_ v / y ]_ [_ u / x ]_ B ) |
19 |
18
|
sumeq2sdv |
|- ( ( x = u /\ y = v ) -> sum_ k e. A B = sum_ k e. A [_ v / y ]_ [_ u / x ]_ B ) |
20 |
6 7 12 15 19
|
cbvmpo |
|- ( x e. X , y e. Y |-> sum_ k e. A B ) = ( u e. X , v e. Y |-> sum_ k e. A [_ v / y ]_ [_ u / x ]_ B ) |
21 |
|
vex |
|- u e. _V |
22 |
|
vex |
|- v e. _V |
23 |
21 22
|
op2ndd |
|- ( z = <. u , v >. -> ( 2nd ` z ) = v ) |
24 |
23
|
csbeq1d |
|- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B = [_ v / y ]_ [_ ( 1st ` z ) / x ]_ B ) |
25 |
21 22
|
op1std |
|- ( z = <. u , v >. -> ( 1st ` z ) = u ) |
26 |
25
|
csbeq1d |
|- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ B = [_ u / x ]_ B ) |
27 |
26
|
csbeq2dv |
|- ( z = <. u , v >. -> [_ v / y ]_ [_ ( 1st ` z ) / x ]_ B = [_ v / y ]_ [_ u / x ]_ B ) |
28 |
24 27
|
eqtrd |
|- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B = [_ v / y ]_ [_ u / x ]_ B ) |
29 |
28
|
sumeq2sdv |
|- ( z = <. u , v >. -> sum_ k e. A [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B = sum_ k e. A [_ v / y ]_ [_ u / x ]_ B ) |
30 |
29
|
mpompt |
|- ( z e. ( X X. Y ) |-> sum_ k e. A [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B ) = ( u e. X , v e. Y |-> sum_ k e. A [_ v / y ]_ [_ u / x ]_ B ) |
31 |
20 30
|
eqtr4i |
|- ( x e. X , y e. Y |-> sum_ k e. A B ) = ( z e. ( X X. Y ) |-> sum_ k e. A [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B ) |
32 |
|
txtopon |
|- ( ( J e. ( TopOn ` X ) /\ L e. ( TopOn ` Y ) ) -> ( J tX L ) e. ( TopOn ` ( X X. Y ) ) ) |
33 |
2 4 32
|
syl2anc |
|- ( ph -> ( J tX L ) e. ( TopOn ` ( X X. Y ) ) ) |
34 |
|
nfcv |
|- F/_ u B |
35 |
|
nfcv |
|- F/_ v B |
36 |
34 35 11 14 18
|
cbvmpo |
|- ( x e. X , y e. Y |-> B ) = ( u e. X , v e. Y |-> [_ v / y ]_ [_ u / x ]_ B ) |
37 |
28
|
mpompt |
|- ( z e. ( X X. Y ) |-> [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B ) = ( u e. X , v e. Y |-> [_ v / y ]_ [_ u / x ]_ B ) |
38 |
36 37
|
eqtr4i |
|- ( x e. X , y e. Y |-> B ) = ( z e. ( X X. Y ) |-> [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B ) |
39 |
38 5
|
eqeltrrid |
|- ( ( ph /\ k e. A ) -> ( z e. ( X X. Y ) |-> [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B ) e. ( ( J tX L ) Cn K ) ) |
40 |
1 33 3 39
|
fsumcn |
|- ( ph -> ( z e. ( X X. Y ) |-> sum_ k e. A [_ ( 2nd ` z ) / y ]_ [_ ( 1st ` z ) / x ]_ B ) e. ( ( J tX L ) Cn K ) ) |
41 |
31 40
|
eqeltrid |
|- ( ph -> ( x e. X , y e. Y |-> sum_ k e. A B ) e. ( ( J tX L ) Cn K ) ) |