Step |
Hyp |
Ref |
Expression |
1 |
|
mhmcoaddpsr.p |
|- P = ( I mPwSer R ) |
2 |
|
mhmcoaddpsr.q |
|- Q = ( I mPwSer S ) |
3 |
|
mhmcoaddpsr.b |
|- B = ( Base ` P ) |
4 |
|
mhmcoaddpsr.c |
|- C = ( Base ` Q ) |
5 |
|
mhmcoaddpsr.1 |
|- .+ = ( +g ` P ) |
6 |
|
mhmcoaddpsr.2 |
|- .+b = ( +g ` Q ) |
7 |
|
mhmcoaddpsr.h |
|- ( ph -> H e. ( R MndHom S ) ) |
8 |
|
mhmcoaddpsr.f |
|- ( ph -> F e. B ) |
9 |
|
mhmcoaddpsr.g |
|- ( ph -> G e. B ) |
10 |
|
fvexd |
|- ( ph -> ( Base ` R ) e. _V ) |
11 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
12 |
11
|
rabex |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } e. _V |
13 |
12
|
a1i |
|- ( ph -> { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } e. _V ) |
14 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
15 |
|
eqid |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
16 |
1 14 15 3 8
|
psrelbas |
|- ( ph -> F : { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } --> ( Base ` R ) ) |
17 |
10 13 16
|
elmapdd |
|- ( ph -> F e. ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
18 |
1 14 15 3 9
|
psrelbas |
|- ( ph -> G : { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } --> ( Base ` R ) ) |
19 |
10 13 18
|
elmapdd |
|- ( ph -> G e. ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
20 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
21 |
|
eqid |
|- ( +g ` S ) = ( +g ` S ) |
22 |
14 20 21
|
mhmvlin |
|- ( ( H e. ( R MndHom S ) /\ F e. ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) /\ G e. ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) -> ( H o. ( F oF ( +g ` R ) G ) ) = ( ( H o. F ) oF ( +g ` S ) ( H o. G ) ) ) |
23 |
7 17 19 22
|
syl3anc |
|- ( ph -> ( H o. ( F oF ( +g ` R ) G ) ) = ( ( H o. F ) oF ( +g ` S ) ( H o. G ) ) ) |
24 |
1 3 20 5 8 9
|
psradd |
|- ( ph -> ( F .+ G ) = ( F oF ( +g ` R ) G ) ) |
25 |
24
|
coeq2d |
|- ( ph -> ( H o. ( F .+ G ) ) = ( H o. ( F oF ( +g ` R ) G ) ) ) |
26 |
1 2 3 4 7 8
|
mhmcopsr |
|- ( ph -> ( H o. F ) e. C ) |
27 |
1 2 3 4 7 9
|
mhmcopsr |
|- ( ph -> ( H o. G ) e. C ) |
28 |
2 4 21 6 26 27
|
psradd |
|- ( ph -> ( ( H o. F ) .+b ( H o. G ) ) = ( ( H o. F ) oF ( +g ` S ) ( H o. G ) ) ) |
29 |
23 25 28
|
3eqtr4d |
|- ( ph -> ( H o. ( F .+ G ) ) = ( ( H o. F ) .+b ( H o. G ) ) ) |