Theorem pwsinvg 13817

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 2232	. . . 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 13358	. . . . . 6 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
4	3	slotex 13231	. . . . 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 5565	. . . . 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 2232	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
9		eqid 2232	. . . 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 2232	. . . . . . . . 9 |- (Scalar ` R ) = (Scalar ` R )
14	12, 13	pwsval 13496	. . . . . . . 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 5673	. . . . . 6 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
17	11, 16	eqtrid 2277	. . . . 5 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
18	10, 17	eleqtrd 2311	. . . 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 13815	. . 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 5897	. . . . . . . 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 5673	. . . . . 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 2283	. . . . 5 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( invg ` ( ( I X. { R } ) ` x ) ) = M )
26	25	fveq1d 5671	. . . 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 4199	. . 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 2265	. 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 5673	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( invg ` Y ) = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
31	29, 30	eqtrid 2277	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> N = ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
32	31	fveq1d 5671	. 2 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( ( invg ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ` X ) )
33		eqid 2232	. . . . 5 |- ( Base ` R ) = ( Base ` R )
34	12, 33, 11, 20, 2, 10	pwselbas 13499	. . . 4 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X : I --> ( Base ` R ) )
35	34	ffvelcdmda 5811	. . 3 |- ( ( ( R e. Grp /\ I e. V /\ X e. B ) /\ x e. I ) -> ( X ` x ) e. ( Base ` R ) )
36	34	feqmptd 5729	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> X = ( x e. I |-> ( X ` x ) ) )
37	33, 24	grpinvf 13752	. . . . 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 5729	. . 3 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> M = ( y e. ( Base ` R ) |-> ( M ` y ) ) )
40		fveq2 5669	. . 3 |- ( y = ( X ` x ) -> ( M ` y ) = ( M ` ( X ` x ) ) )
41	35, 36, 39, 40	fmptco 5842	. 2 |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( M o. X ) = ( x e. I |-> ( M ` ( X ` x ) ) ) )
42	28, 32, 41	3eqtr4d 2275	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 2203   _Vcvv 2812   {csn 3688   |-> cmpt 4170   X. cxp 4746   o. ccom 4752   -->wf 5347   ` cfv 5351  (class class class)co 6049   Basecbs 13204  Scalarcsca 13285   X__scprds 13470   ^_s cpws 13471   Grpcgrp 13705   invgcminusg 13706

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-map 6883  df-ixp 6933  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-z 9577  df-dec 9709  df-uz 9853  df-fz 10342  df-struct 13206  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-mulr 13296  df-sca 13298  df-vsca 13299  df-ip 13300  df-tset 13301  df-ple 13302  df-ds 13304  df-hom 13306  df-cco 13307  df-rest 13446  df-topn 13447  df-0g 13463  df-topgen 13465  df-pt 13466  df-prds 13472  df-pws 13495  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709

This theorem is referenced by:  pwssub  13818