Theorem mplsubgfileminv 15074

Description: Lemma for mplsubgfi 15075. 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 15070	. . . . 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 15061	. . 3 |- ( ph -> ( N ` X ) = ( ( invg ` R ) o. X ) )
14	1, 2, 3, 4, 5, 6, 12	psrnegcl 15057	. . 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 15066	. . . . . 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 5695	. . . . . . . . 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 15050	. . . . . . . . . 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 5773	. . . . . . . . 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 5697	. . . . . . . . 9 |- ( ( ( ph /\ b e. ( NN0 ^m I ) ) /\ ( X ` b ) = ( 0g ` R ) ) -> ( ( invg ` R ) ` ( X ` b ) ) = ( ( invg ` R ) ` ( 0g ` R ) ) )
31	16, 5	grpinvid 13848	. . . . . . . . . . 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 15066	. . 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   class class class wbr 4128   `'ccnv 4771   "cima 4775   o. ccom 4776   -->wf 5371   ` cfv 5375  (class class class)co 6079   ^m cmap 6916   Fincfn 7016   < clt 8354   NNcn 9287   NN0cn0 9546   Basecbs 13335   0gc0g 13593   Grpcgrp 13788   invgcminusg 13789   mPwSer cmps 15028   mPoly cmpl 15029

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem 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 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-1st 6368  df-2nd 6369  df-1o 6681  df-er 6801  df-map 6918  df-ixp 6975  df-en 7017  df-fin 7019  df-sup 7318  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-fz 10395  df-struct 13337  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-ip 13432  df-tset 13433  df-ple 13434  df-ds 13436  df-hom 13438  df-cco 13439  df-rest 13578  df-topn 13579  df-0g 13595  df-topgen 13597  df-pt 13598  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-prds 14153  df-pws 14186  df-psr 15030  df-mplcoe 15031

This theorem is referenced by:  mplsubgfi  15075