Theorem pwsinvg 13700

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 2231	. . . 4 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
2		simp2 1024	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> I e. V )
3		scaslid 13241	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
4	3	slotex 13114	. . . . 5 |- ( R e. Grp -> (Scalar ` R ) e. _V )
5	4	3ad2ant1 1044	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> (Scalar ` R ) e. _V )
6		fconst6g 5535	. . . . 5 |- ( R e. Grp -> ( I X. { R } ) : I --> Grp )
7	6	3ad2ant1 1044	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( I X. { R } ) : I --> Grp )
8		eqid 2231	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
9		eqid 2231	. . . 4 |- ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
10		simp3 1025	. . . . 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 2231	. . . . . . . . 9 |- (Scalar ` R ) = (Scalar ` R )
14	12, 13	pwsval 13379	. . . . . . . 8 |- ( ( R e. Grp /\ I e. V ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
15	14	3adant3 1043	. . . . . . 7 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
16	15	fveq2d 5643	. . . . . 6 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
17	11, 16	eqtrid 2276	. . . . 5 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
18	10, 17	eleqtrd 2310	. . . 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 13698	. . 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 1023	. . . . . . . 8 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> R e. Grp )
21		fvconst2g 5868	. . . . . . . 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 5643	. . . . . 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 2282	. . . . 5 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( invg ` ( ( I X. { R } ) ` x ) ) = M )
26	25	fveq1d 5641	. . . 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 4179	. . 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 2264	. 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 5643	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( invg ` Y ) = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
31	29, 30	eqtrid 2276	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> N = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
32	31	fveq1d 5641	. 2 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ` X ) )
33		eqid 2231	. . . . 5 |- ( Base ` R ) = ( Base ` R )
34	12, 33, 11, 20, 2, 10	pwselbas 13382	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X : I --> ( Base ` R ) )
35	34	ffvelcdmda 5782	. . 3 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( X ` x ) e. ( Base ` R ) )
36	34	feqmptd 5699	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X = ( x e. I |-> ( X ` x ) ) )
37	33, 24	grpinvf 13635	. . . . 5 |- ( R e. Grp -> M : ( Base ` R ) --> ( Base ` R ) )
38	37	3ad2ant1 1044	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> M : ( Base ` R ) --> ( Base ` R ) )
39	38	feqmptd 5699	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> M = ( y e. ( Base ` R ) |-> ( M ` y ) ) )
40		fveq2 5639	. . 3 |- ( y = ( X ` x ) -> ( M ` y ) = ( M ` ( X ` x ) ) )
41	35, 36, 39, 40	fmptco 5813	. 2 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( M o. X ) = ( x e. I |-> ( M ` ( X ` x ) ) ) )
42	28, 32, 41	3eqtr4d 2274	1 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( M o. X ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   |-> cmpt 4150   X. cxp 4723   o. ccom 4729   -->wf 5322   ` cfv 5326  (class class class)co 6018   Basecbs 13087  Scalarcsca 13168   X__scprds 13353   ^_s cpws 13354   Grpcgrp 13588   invgcminusg 13589

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-ixp 6868  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-struct 13089  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-mulr 13179  df-sca 13181  df-vsca 13182  df-ip 13183  df-tset 13184  df-ple 13185  df-ds 13187  df-hom 13189  df-cco 13190  df-rest 13329  df-topn 13330  df-0g 13346  df-topgen 13348  df-pt 13349  df-prds 13355  df-pws 13378  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-minusg 13592

This theorem is referenced by:  pwssub  13701