Theorem mplsubgfileminv 15093

Description: Lemma for mplsubgfi 15094. 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 2238	. . . 4 |- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
5		eqid 2238	. . . 4 |- ( invg ` R ) = ( invg ` R )
6		eqid 2238	. . . 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 15089	. . . . 5 |- U C_ ( Base ` S )
11		mplsubgfileminv.x	. . . . 5 |- ( ph -> X e. U )
12	10, 11	sselid 3246	. . . 4 |- ( ph -> X e. ( Base ` S ) )
13	1, 2, 3, 4, 5, 6, 7, 12	psrneg 15080	. . 3 |- ( ph -> ( N ` X ) = ( ( invg ` R ) o. X ) )
14	1, 2, 3, 4, 5, 6, 12	psrnegcl 15076	. . 3 |- ( ph -> ( ( invg ` R ) o. X ) e. ( Base ` S ) )
15	13, 14	eqeltrd 2315	. 2 |- ( ph -> ( N ` X ) e. ( Base ` S ) )
16		eqid 2238	. . . . . . 7 |- ( 0g ` R ) = ( 0g ` R )
17	8, 1, 6, 16, 9	mplelbascoe 15085	. . . . . 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 415	. . . . 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 5697	. . . . . . . . 9 |- ( ph -> ( ( N ` X ) ` b ) = ( ( ( invg ` R ) o. X ) ` b ) )
22	21	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( N ` X ) ` b ) = ( ( ( invg ` R ) o. X ) ` b ) )
23		eqid 2238	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
24	1, 23, 2, 6, 12	psrelbasfi 15069	. . . . . . . . . 10 |- ( ph -> X : ( NN0 ^m I ) --> ( Base ` R ) )
25	24	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> X : ( NN0 ^m I ) --> ( Base ` R ) )
26		simplr 533	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> b e. ( NN0 ^m I ) )
27		fvco3 5776	. . . . . . . . 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 415	. . . . . . . 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 5699	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( invg ` R ) ` ( X ` b ) ) = ( ( invg ` R ) ` ( 0g ` R ) ) )
31	16, 5	grpinvid 13867	. . . . . . . . . . 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 492	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( invg ` R ) ` ( 0g ` R ) ) = ( 0g ` R ) )
34	30, 33	eqtrd 2271	. . . . . . . 8 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( invg ` R ) ` ( X ` b ) ) = ( 0g ` R ) )
35	22, 28, 34	3eqtrd 2275	. . . . . . 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 2617	. . . 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 2651	. . 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 15085	. . 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 415	. 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 957	1 |- ( ph -> ( N ` X ) e. U )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   class class class wbr 4130   `'ccnv 4773   "cima 4777   o. ccom 4778   -->wf 5373   ` cfv 5377  (class class class)co 6085   ^m cmap 6922   Fincfn 7022   < clt 8360   NNcn 9305   NN0cn0 9565   Basecbs 13354   0gc0g 13612   Grpcgrp 13807   invgcminusg 13808   mPwSer cmps 15047   mPoly cmpl 15048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-1o 6687  df-er 6807  df-map 6924  df-ixp 6981  df-en 7023  df-fin 7025  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-z 9647  df-dec 9780  df-uz 9924  df-fz 10414  df-struct 13356  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-sca 13449  df-vsca 13450  df-ip 13451  df-tset 13452  df-ple 13453  df-ds 13455  df-hom 13457  df-cco 13458  df-rest 13597  df-topn 13598  df-0g 13614  df-topgen 13616  df-pt 13617  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-prds 14172  df-pws 14205  df-psr 15049  df-mplcoe 15050

This theorem is used by:  mplsubgfi  15094