Theorem pwsinvg 13909

Description: Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)

Hypotheses

Ref	Expression
pwsgrp.y	|- Y = ( R ^s I )
pwsinvg.b	|- B = ( Base ` Y )
pwsinvg.m	|- M = ( invg ` R )
pwsinvg.n	|- N = ( invg ` Y )

Assertion

Ref	Expression
pwsinvg	|- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( M o. X ) )

Proof of Theorem pwsinvg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . . 4 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
2		simp2 1025	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> I e. V )
3		scaslid 13450	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
4	3	slotex 13323	. . . . 5 |- ( R e. Grp -> (Scalar ` R ) e. _V )
5	4	3ad2ant1 1045	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> (Scalar ` R ) e. _V )
6		fconst6g 5571	. . . . 5 |- ( R e. Grp -> ( I X. { R } ) : I --> Grp )
7	6	3ad2ant1 1045	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( I X. { R } ) : I --> Grp )
8		eqid 2234	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
9		eqid 2234	. . . 4 |- ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
10		simp3 1026	. . . . 5 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X e. B )
11		pwsinvg.b	. . . . . 6 |- B = ( Base ` Y )
12		pwsgrp.y	. . . . . . . . 9 |- Y = ( R ^s I )
13		eqid 2234	. . . . . . . . 9 |- (Scalar ` R ) = (Scalar ` R )
14	12, 13	pwsval 13588	. . . . . . . 8 |- ( ( R e. Grp /\ I e. V ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
15	14	3adant3 1044	. . . . . . 7 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
16	15	fveq2d 5679	. . . . . 6 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
17	11, 16	eqtrid 2279	. . . . 5 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
18	10, 17	eleqtrd 2313	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X e. ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
19	1, 2, 5, 7, 8, 9, 18	prdsinvgd 13907	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ` X ) = ( x e. I |-> ( ( invg ` ( ( I X. { R } ) ` x ) ) ` ( X ` x ) ) ) )
20		simp1 1024	. . . . . . . 8 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> R e. Grp )
21		fvconst2g 5903	. . . . . . . 8 |- ( ( R e. Grp /\ x e. I ) -> ( ( I X. { R } ) ` x ) = R )
22	20, 21	sylan 283	. . . . . . 7 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( ( I X. { R } ) ` x ) = R )
23	22	fveq2d 5679	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( invg ` ( ( I X. { R } ) ` x ) ) = ( invg ` R ) )
24		pwsinvg.m	. . . . . 6 |- M = ( invg ` R )
25	23, 24	eqtr4di 2285	. . . . 5 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( invg ` ( ( I X. { R } ) ` x ) ) = M )
26	25	fveq1d 5677	. . . 4 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( ( invg ` ( ( I X. { R } ) ` x ) ) ` ( X ` x ) ) = ( M ` ( X ` x ) ) )
27	26	mpteq2dva 4205	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( x e. I |-> ( ( invg ` ( ( I X. { R } ) ` x ) ) ` ( X ` x ) ) ) = ( x e. I |-> ( M ` ( X ` x ) ) ) )
28	19, 27	eqtrd 2267	. 2 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ` X ) = ( x e. I |-> ( M ` ( X ` x ) ) ) )
29		pwsinvg.n	. . . 4 |- N = ( invg ` Y )
30	15	fveq2d 5679	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( invg ` Y ) = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
31	29, 30	eqtrid 2279	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> N = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
32	31	fveq1d 5677	. 2 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ` X ) )
33		eqid 2234	. . . . 5 |- ( Base ` R ) = ( Base ` R )
34	12, 33, 11, 20, 2, 10	pwselbas 13591	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X : I --> ( Base ` R ) )
35	34	ffvelcdmda 5817	. . 3 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( X ` x ) e. ( Base ` R ) )
36	34	feqmptd 5735	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X = ( x e. I |-> ( X ` x ) ) )
37	33, 24	grpinvf 13844	. . . . 5 |- ( R e. Grp -> M : ( Base ` R ) --> ( Base ` R ) )
38	37	3ad2ant1 1045	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> M : ( Base ` R ) --> ( Base ` R ) )
39	38	feqmptd 5735	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> M = ( y e. ( Base ` R ) |-> ( M ` y ) ) )
40		fveq2 5675	. . 3 |- ( y = ( X ` x ) -> ( M ` y ) = ( M ` ( X ` x ) ) )
41	35, 36, 39, 40	fmptco 5848	. 2 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( M o. X ) = ( x e. I |-> ( M ` ( X ` x ) ) ) )
42	28, 32, 41	3eqtr4d 2277	1 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( M o. X ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3694   |-> cmpt 4176   X. cxp 4752   o. ccom 4758   -->wf 5353   ` cfv 5357  (class class class)co 6058   Basecbs 13296  Scalarcsca 13377   X__scprds 13562   ^_s cpws 13563   Grpcgrp 13797   invgcminusg 13798

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-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  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-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-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-id 4419  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-1st 6347  df-2nd 6348  df-map 6897  df-ixp 6947  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-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

This theorem is referenced by:  pwssub  13910