Theorem mplsubgfileminv 14967

Description: Lemma for mplsubgfi 14968. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)

Hypotheses

Ref	Expression
mplsubg.s	|- S = ( I mPwSer R )
mplsubg.p	|- P = ( I mPoly R )
mplsubg.u	|- U = ( Base ` P )
mplsubg.i	|- ( ph -> I e. Fin )
mplsubg.r	|- ( ph -> R e. Grp )
mplsubgfileminv.x	|- ( ph -> X e. U )
mplsubgfileminv.inv	|- N = ( invg ` S )

Assertion

Ref	Expression
mplsubgfileminv	|- ( ph -> ( N ` X ) e. U )

Proof of Theorem mplsubgfileminv

Dummy variables a b k f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplsubg.s	. . . 4 |- S = ( I mPwSer R )
2		mplsubg.i	. . . 4 |- ( ph -> I e. Fin )
3		mplsubg.r	. . . 4 |- ( ph -> R e. Grp )
4		eqid 2234	. . . 4 |- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
5		eqid 2234	. . . 4 |- ( invg ` R ) = ( invg ` R )
6		eqid 2234	. . . 4 |- ( Base ` S ) = ( Base ` S )
7		mplsubgfileminv.inv	. . . 4 |- N = ( invg ` S )
8		mplsubg.p	. . . . . 6 |- P = ( I mPoly R )
9		mplsubg.u	. . . . . 6 |- U = ( Base ` P )
10	8, 1, 9, 6	mplbasss 14963	. . . . 5 |- U C_ ( Base ` S )
11		mplsubgfileminv.x	. . . . 5 |- ( ph -> X e. U )
12	10, 11	sselid 3240	. . . 4 |- ( ph -> X e. ( Base ` S ) )
13	1, 2, 3, 4, 5, 6, 7, 12	psrneg 14954	. . 3 |- ( ph -> ( N ` X ) = ( ( invg ` R ) o. X ) )
14	1, 2, 3, 4, 5, 6, 12	psrnegcl 14950	. . 3 |- ( ph -> ( ( invg ` R ) o. X ) e. ( Base ` S ) )
15	13, 14	eqeltrd 2311	. 2 |- ( ph -> ( N ` X ) e. ( Base ` S ) )
16		eqid 2234	. . . . . . 7 |- ( 0g ` R ) = ( 0g ` R )
17	8, 1, 6, 16, 9	mplelbascoe 14959	. . . . . 6 |- ( ( I e. Fin /\ R e. Grp ) -> ( X e. U <-> ( X e. ( Base ` S ) /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = ( 0g ` R ) ) ) ) )
18	2, 3, 17	syl2anc 411	. . . . 5 |- ( ph -> ( X e. U <-> ( X e. ( Base ` S ) /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = ( 0g ` R ) ) ) ) )
19	11, 18	mpbid 147	. . . 4 |- ( ph -> ( X e. ( Base ` S ) /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = ( 0g ` R ) ) ) )
20	19	simprd 114	. . 3 |- ( ph -> E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = ( 0g ` R ) ) )
21	13	fveq1d 5677	. . . . . . . . 9 |- ( ph -> ( ( N ` X ) ` b ) = ( ( ( invg ` R ) o. X ) ` b ) )
22	21	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( N ` X ) ` b ) = ( ( ( invg ` R ) o. X ) ` b ) )
23		eqid 2234	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
24	1, 23, 2, 6, 12	psrelbasfi 14943	. . . . . . . . . 10 |- ( ph -> X : ( NN0 ^m I ) --> ( Base ` R ) )
25	24	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> X : ( NN0 ^m I ) --> ( Base ` R ) )
26		simplr 529	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> b e. ( NN0 ^m I ) )
27		fvco3 5753	. . . . . . . . 9 |- ( ( X : ( NN0 ^m I ) --> ( Base ` R ) /\ b e. ( NN0 ^m I ) ) -> ( ( ( invg ` R ) o. X ) ` b ) = ( ( invg ` R ) ` ( X ` b ) ) )
28	25, 26, 27	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( ( invg ` R ) o. X ) ` b ) = ( ( invg ` R ) ` ( X ` b ) ) )
29		simpr 110	. . . . . . . . . 10 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( X ` b ) = ( 0g ` R ) )
30	29	fveq2d 5679	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( invg ` R ) ` ( X ` b ) ) = ( ( invg ` R ) ` ( 0g ` R ) ) )
31	16, 5	grpinvid 13857	. . . . . . . . . . 11 |- ( R e. Grp -> ( ( invg ` R ) ` ( 0g ` R ) ) = ( 0g ` R ) )
32	3, 31	syl 14	. . . . . . . . . 10 |- ( ph -> ( ( invg ` R ) ` ( 0g ` R ) ) = ( 0g ` R ) )
33	32	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( invg ` R ) ` ( 0g ` R ) ) = ( 0g ` R ) )
34	30, 33	eqtrd 2267	. . . . . . . 8 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( invg ` R ) ` ( X ` b ) ) = ( 0g ` R ) )
35	22, 28, 34	3eqtrd 2271	. . . . . . 7 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) )
36	35	ex 115	. . . . . 6 |- ( ( ph /\ b e. ( NN0 ^m I ) ) -> ( ( X ` b ) = ( 0g ` R ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) ) )
37	36	imim2d 54	. . . . 5 |- ( ( ph /\ b e. ( NN0 ^m I ) ) -> ( ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = ( 0g ` R ) ) -> ( A. k e. I ( a ` k ) < ( b ` k ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) ) ) )
38	37	ralimdva 2611	. . . 4 |- ( ph -> ( A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = ( 0g ` R ) ) -> A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) ) ) )
39	38	reximdv 2645	. . 3 |- ( ph -> ( E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = ( 0g ` R ) ) -> E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) ) ) )
40	20, 39	mpd 13	. 2 |- ( ph -> E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) ) )
41	8, 1, 6, 16, 9	mplelbascoe 14959	. . 3 |- ( ( I e. Fin /\ R e. Grp ) -> ( ( N ` X ) e. U <-> ( ( N ` X ) e. ( Base ` S ) /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) ) ) ) )
42	2, 3, 41	syl2anc 411	. 2 |- ( ph -> ( ( N ` X ) e. U <-> ( ( N ` X ) e. ( Base ` S ) /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( ( N ` X ) ` b ) = ( 0g ` R ) ) ) ) )
43	15, 40, 42	mpbir2and 953	1 |- ( ph -> ( N ` X ) e. U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   class class class wbr 4114   `'ccnv 4753   "cima 4757   o. ccom 4758   -->wf 5353   ` cfv 5357  (class class class)co 6058   ^m cmap 6895   Fincfn 6988   < clt 8324   NNcn 9254   NN0cn0 9513   Basecbs 13296   0gc0g 13553   Grpcgrp 13797   invgcminusg 13798   mPwSer cmps 14921   mPoly cmpl 14922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-1o 6660  df-er 6780  df-map 6897  df-ixp 6947  df-en 6989  df-fin 6991  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-ip 13392  df-tset 13393  df-ple 13394  df-ds 13396  df-hom 13398  df-cco 13399  df-rest 13538  df-topn 13539  df-0g 13555  df-topgen 13557  df-pt 13558  df-prds 13564  df-pws 13587  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801  df-psr 14923  df-mplcoe 14924

This theorem is referenced by:  mplsubgfi  14968