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Metamath Proof Explorer

Theorem mhpaddcl

Description: Homogeneous polynomials are closed under addition. (Contributed by SN, 26-Aug-2023) Remove closure hypotheses. (Revised by SN, 4-Sep-2025)

Ref Expression
Hypotheses mhpaddcl.h 
|- H = ( I mHomP R )
mhpaddcl.p 
|- P = ( I mPoly R )
mhpaddcl.a 
|- .+ = ( +g ` P )
mhpaddcl.r 
|- ( ph -> R e. Grp )
mhpaddcl.x 
|- ( ph -> X e. ( H ` N ) )
mhpaddcl.y 
|- ( ph -> Y e. ( H ` N ) )
Assertion mhpaddcl 
|- ( ph -> ( X .+ Y ) e. ( H ` N ) )

Proof

Step Hyp Ref Expression
1 mhpaddcl.h 
 |-  H = ( I mHomP R )
2 mhpaddcl.p 
 |-  P = ( I mPoly R )
3 mhpaddcl.a 
 |-  .+ = ( +g ` P )
4 mhpaddcl.r 
 |-  ( ph -> R e. Grp )
5 mhpaddcl.x 
 |-  ( ph -> X e. ( H ` N ) )
6 mhpaddcl.y 
 |-  ( ph -> Y e. ( H ` N ) )
7 eqid 
 |-  ( Base ` P ) = ( Base ` P )
8 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
9 eqid 
 |-  { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
10 1 5 mhprcl 
 |-  ( ph -> N e. NN0 )
11 reldmmhp 
 |-  Rel dom mHomP
12 11 1 5 elfvov1 
 |-  ( ph -> I e. _V )
13 2 mplgrp 
 |-  ( ( I e. _V /\ R e. Grp ) -> P e. Grp )
14 12 4 13 syl2anc 
 |-  ( ph -> P e. Grp )
15 1 2 7 5 mhpmpl 
 |-  ( ph -> X e. ( Base ` P ) )
16 1 2 7 6 mhpmpl 
 |-  ( ph -> Y e. ( Base ` P ) )
17 7 3 14 15 16 grpcld 
 |-  ( ph -> ( X .+ Y ) e. ( Base ` P ) )
18 eqid 
 |-  ( +g ` R ) = ( +g ` R )
19 2 7 18 3 15 16 mpladd 
 |-  ( ph -> ( X .+ Y ) = ( X oF ( +g ` R ) Y ) )
20 19 oveq1d 
 |-  ( ph -> ( ( X .+ Y ) supp ( 0g ` R ) ) = ( ( X oF ( +g ` R ) Y ) supp ( 0g ` R ) ) )
21 ovexd 
 |-  ( ph -> ( NN0 ^m I ) e. _V )
22 9 21 rabexd 
 |-  ( ph -> { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } e. _V )
23 eqid 
 |-  ( Base ` R ) = ( Base ` R )
24 23 8 grpidcl 
 |-  ( R e. Grp -> ( 0g ` R ) e. ( Base ` R ) )
25 4 24 syl 
 |-  ( ph -> ( 0g ` R ) e. ( Base ` R ) )
26 2 23 7 9 15 mplelf 
 |-  ( ph -> X : { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } --> ( Base ` R ) )
27 2 23 7 9 16 mplelf 
 |-  ( ph -> Y : { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } --> ( Base ` R ) )
28 23 18 8 4 25 grplidd 
 |-  ( ph -> ( ( 0g ` R ) ( +g ` R ) ( 0g ` R ) ) = ( 0g ` R ) )
29 22 25 26 27 28 suppofssd 
 |-  ( ph -> ( ( X oF ( +g ` R ) Y ) supp ( 0g ` R ) ) C_ ( ( X supp ( 0g ` R ) ) u. ( Y supp ( 0g ` R ) ) ) )
30 20 29 eqsstrd 
 |-  ( ph -> ( ( X .+ Y ) supp ( 0g ` R ) ) C_ ( ( X supp ( 0g ` R ) ) u. ( Y supp ( 0g ` R ) ) ) )
31 1 8 9 5 mhpdeg 
 |-  ( ph -> ( X supp ( 0g ` R ) ) C_ { g e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } | ( ( CCfld |`s NN0 ) gsum g ) = N } )
32 1 8 9 6 mhpdeg 
 |-  ( ph -> ( Y supp ( 0g ` R ) ) C_ { g e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } | ( ( CCfld |`s NN0 ) gsum g ) = N } )
33 31 32 unssd 
 |-  ( ph -> ( ( X supp ( 0g ` R ) ) u. ( Y supp ( 0g ` R ) ) ) C_ { g e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } | ( ( CCfld |`s NN0 ) gsum g ) = N } )
34 30 33 sstrd 
 |-  ( ph -> ( ( X .+ Y ) supp ( 0g ` R ) ) C_ { g e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } | ( ( CCfld |`s NN0 ) gsum g ) = N } )
35 1 2 7 8 9 10 17 34 ismhp2 
 |-  ( ph -> ( X .+ Y ) e. ( H ` N ) )