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Metamath Proof Explorer

Theorem mplgsum

Description: Finite commutative sums of polynomials are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026)

Ref Expression
Hypotheses mplgsum.p 
|- P = ( I mPoly R )
mplgsum.b 
|- B = ( Base ` P )
mplgsum.r 
|- ( ph -> R e. Ring )
mplgsum.i 
|- ( ph -> I e. V )
mplgsum.d 
|- D = { h e. ( NN0 ^m I ) | h finSupp 0 }
mplgsum.a 
|- ( ph -> A e. Fin )
mplgsum.f 
|- ( ph -> F : A --> B )
Assertion mplgsum 
|- ( ph -> ( P gsum F ) = ( y e. D |-> ( R gsum ( k e. A |-> ( ( F ` k ) ` y ) ) ) ) )

Proof

Step Hyp Ref Expression
1 mplgsum.p 
 |-  P = ( I mPoly R )
2 mplgsum.b 
 |-  B = ( Base ` P )
3 mplgsum.r 
 |-  ( ph -> R e. Ring )
4 mplgsum.i 
 |-  ( ph -> I e. V )
5 mplgsum.d 
 |-  D = { h e. ( NN0 ^m I ) | h finSupp 0 }
6 mplgsum.a 
 |-  ( ph -> A e. Fin )
7 mplgsum.f 
 |-  ( ph -> F : A --> B )
8 eqid 
 |-  ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) )
9 eqid 
 |-  ( +g ` ( I mPwSer R ) ) = ( +g ` ( I mPwSer R ) )
10 eqid 
 |-  ( I mPwSer R ) = ( I mPwSer R )
11 1 10 2 mplval2 
 |-  P = ( ( I mPwSer R ) |`s B )
12 ovexd 
 |-  ( ph -> ( I mPwSer R ) e. _V )
13 1 10 2 8 mplbasss 
 |-  B C_ ( Base ` ( I mPwSer R ) )
14 13 a1i 
 |-  ( ph -> B C_ ( Base ` ( I mPwSer R ) ) )
15 3 ringgrpd 
 |-  ( ph -> R e. Grp )
16 5 psrbasfsupp 
 |-  D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
17 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
18 eqid 
 |-  ( 0g ` ( I mPwSer R ) ) = ( 0g ` ( I mPwSer R ) )
19 10 4 15 16 17 18 psr0 
 |-  ( ph -> ( 0g ` ( I mPwSer R ) ) = ( D X. { ( 0g ` R ) } ) )
20 eqid 
 |-  ( 0g ` P ) = ( 0g ` P )
21 1 16 17 20 4 15 mpl0 
 |-  ( ph -> ( 0g ` P ) = ( D X. { ( 0g ` R ) } ) )
22 19 21 eqtr4d 
 |-  ( ph -> ( 0g ` ( I mPwSer R ) ) = ( 0g ` P ) )
23 1 mplgrp 
 |-  ( ( I e. V /\ R e. Grp ) -> P e. Grp )
24 4 15 23 syl2anc 
 |-  ( ph -> P e. Grp )
25 2 20 grpidcl 
 |-  ( P e. Grp -> ( 0g ` P ) e. B )
26 24 25 syl 
 |-  ( ph -> ( 0g ` P ) e. B )
27 22 26 eqeltrd 
 |-  ( ph -> ( 0g ` ( I mPwSer R ) ) e. B )
28 10 4 15 psrgrp 
 |-  ( ph -> ( I mPwSer R ) e. Grp )
29 28 adantr 
 |-  ( ( ph /\ x e. ( Base ` ( I mPwSer R ) ) ) -> ( I mPwSer R ) e. Grp )
30 simpr 
 |-  ( ( ph /\ x e. ( Base ` ( I mPwSer R ) ) ) -> x e. ( Base ` ( I mPwSer R ) ) )
31 8 9 18 29 30 grplidd 
 |-  ( ( ph /\ x e. ( Base ` ( I mPwSer R ) ) ) -> ( ( 0g ` ( I mPwSer R ) ) ( +g ` ( I mPwSer R ) ) x ) = x )
32 8 9 18 29 30 grpridd 
 |-  ( ( ph /\ x e. ( Base ` ( I mPwSer R ) ) ) -> ( x ( +g ` ( I mPwSer R ) ) ( 0g ` ( I mPwSer R ) ) ) = x )
33 31 32 jca 
 |-  ( ( ph /\ x e. ( Base ` ( I mPwSer R ) ) ) -> ( ( ( 0g ` ( I mPwSer R ) ) ( +g ` ( I mPwSer R ) ) x ) = x /\ ( x ( +g ` ( I mPwSer R ) ) ( 0g ` ( I mPwSer R ) ) ) = x ) )
34 8 9 11 12 6 14 7 27 33 gsumress 
 |-  ( ph -> ( ( I mPwSer R ) gsum F ) = ( P gsum F ) )
35 7 14 fssd 
 |-  ( ph -> F : A --> ( Base ` ( I mPwSer R ) ) )
36 10 8 3 4 5 6 35 psrgsum 
 |-  ( ph -> ( ( I mPwSer R ) gsum F ) = ( y e. D |-> ( R gsum ( k e. A |-> ( ( F ` k ) ` y ) ) ) ) )
37 34 36 eqtr3d 
 |-  ( ph -> ( P gsum F ) = ( y e. D |-> ( R gsum ( k e. A |-> ( ( F ` k ) ` y ) ) ) ) )