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Metamath Proof Explorer

Theorem mhmcoaddmpl

Description: Show that the ring homomorphism in rhmmpl preserves addition. (Contributed by SN, 8-Feb-2025)

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

Proof

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