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 ) ) )