Step |
Hyp |
Ref |
Expression |
1 |
|
psrvsca.s |
|- S = ( I mPwSer R ) |
2 |
|
psrvsca.n |
|- .xb = ( .s ` S ) |
3 |
|
psrvsca.k |
|- K = ( Base ` R ) |
4 |
|
psrvsca.b |
|- B = ( Base ` S ) |
5 |
|
psrvsca.m |
|- .x. = ( .r ` R ) |
6 |
|
psrvsca.d |
|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
7 |
|
psrvsca.x |
|- ( ph -> X e. K ) |
8 |
|
psrvsca.y |
|- ( ph -> F e. B ) |
9 |
|
psrvscaval.y |
|- ( ph -> Y e. D ) |
10 |
1 2 3 4 5 6 7 8
|
psrvsca |
|- ( ph -> ( X .xb F ) = ( ( D X. { X } ) oF .x. F ) ) |
11 |
10
|
fveq1d |
|- ( ph -> ( ( X .xb F ) ` Y ) = ( ( ( D X. { X } ) oF .x. F ) ` Y ) ) |
12 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
13 |
6 12
|
rabex2 |
|- D e. _V |
14 |
13
|
a1i |
|- ( ph -> D e. _V ) |
15 |
1 3 6 4 8
|
psrelbas |
|- ( ph -> F : D --> K ) |
16 |
15
|
ffnd |
|- ( ph -> F Fn D ) |
17 |
|
eqidd |
|- ( ( ph /\ Y e. D ) -> ( F ` Y ) = ( F ` Y ) ) |
18 |
14 7 16 17
|
ofc1 |
|- ( ( ph /\ Y e. D ) -> ( ( ( D X. { X } ) oF .x. F ) ` Y ) = ( X .x. ( F ` Y ) ) ) |
19 |
9 18
|
mpdan |
|- ( ph -> ( ( ( D X. { X } ) oF .x. F ) ` Y ) = ( X .x. ( F ` Y ) ) ) |
20 |
11 19
|
eqtrd |
|- ( ph -> ( ( X .xb F ) ` Y ) = ( X .x. ( F ` Y ) ) ) |