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Metamath Proof Explorer

Theorem mhmcoaddpsr

Description: Show that the ring homomorphism in rhmpsr preserves addition. (Contributed by SN, 18-May-2025)

Ref Expression
Hypotheses mhmcoaddpsr.p 
|- P = ( I mPwSer R )
mhmcoaddpsr.q 
|- Q = ( I mPwSer S )
mhmcoaddpsr.b 
|- B = ( Base ` P )
mhmcoaddpsr.c 
|- C = ( Base ` Q )
mhmcoaddpsr.1 
|- .+ = ( +g ` P )
mhmcoaddpsr.2 
|- .+b = ( +g ` Q )
mhmcoaddpsr.h 
|- ( ph -> H e. ( R MndHom S ) )
mhmcoaddpsr.f 
|- ( ph -> F e. B )
mhmcoaddpsr.g 
|- ( ph -> G e. B )
Assertion mhmcoaddpsr 
|- ( ph -> ( H o. ( F .+ G ) ) = ( ( H o. F ) .+b ( H o. G ) ) )

Proof

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 ) ) )