Theorem pwssub 13910

Description: Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
pwssub	|- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F .- G ) = ( F oF M G ) )

Proof of Theorem pwssub

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

Step	Hyp	Ref	Expression
1		simplr 529	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> I e. V )
2		pwsgrp.y	. . . . . 6 |- Y = ( R ^s I )
3		eqid 2234	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4		pwsinvg.b	. . . . . 6 |- B = ( Base ` Y )
5		simpll 527	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> R e. Grp )
6		simprl 531	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F e. B )
7	2, 3, 4, 5, 1, 6	pwselbas 13591	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F : I --> ( Base ` R ) )
8	7	ffvelcdmda 5817	. . . 4 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( F ` x ) e. ( Base ` R ) )
9		eqid 2234	. . . . . . . 8 |- ( invg ` R ) = ( invg ` R )
10	3, 9	grpinvf 13844	. . . . . . 7 |- ( R e. Grp -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
11	10	ad2antrr 488	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
12	11	adantr 276	. . . . 5 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
13		simprr 533	. . . . . . 7 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G e. B )
14	2, 3, 4, 5, 1, 13	pwselbas 13591	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G : I --> ( Base ` R ) )
15	14	ffvelcdmda 5817	. . . . 5 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( G ` x ) e. ( Base ` R ) )
16	12, 15	ffvelcdmd 5818	. . . 4 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( ( invg ` R ) ` ( G ` x ) ) e. ( Base ` R ) )
17	7	feqmptd 5735	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F = ( x e. I |-> ( F ` x ) ) )
18		eqid 2234	. . . . . . 7 |- ( invg ` Y ) = ( invg ` Y )
19	2, 4, 9, 18	pwsinvg 13909	. . . . . 6 |- ( ( R e. Grp /\ I e. V /\ G e. B ) -> ( ( invg ` Y ) ` G ) = ( ( invg ` R ) o. G ) )
20	5, 1, 13, 19	syl3anc 1274	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` Y ) ` G ) = ( ( invg ` R ) o. G ) )
21	14	feqmptd 5735	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G = ( x e. I |-> ( G ` x ) ) )
22	11	feqmptd 5735	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( invg ` R ) = ( y e. ( Base ` R ) |-> ( ( invg ` R ) ` y ) ) )
23		fveq2 5675	. . . . . 6 |- ( y = ( G ` x ) -> ( ( invg ` R ) ` y ) = ( ( invg ` R ) ` ( G ` x ) ) )
24	15, 21, 22, 23	fmptco 5848	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` R ) o. G ) = ( x e. I |-> ( ( invg ` R ) ` ( G ` x ) ) ) )
25	20, 24	eqtrd 2267	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` Y ) ` G ) = ( x e. I |-> ( ( invg ` R ) ` ( G ` x ) ) ) )
26	1, 8, 16, 17, 25	offval2 6291	. . 3 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F oF ( +g ` R ) ( ( invg ` Y ) ` G ) ) = ( x e. I |-> ( ( F ` x ) ( +g ` R ) ( ( invg ` R ) ` ( G ` x ) ) ) ) )
27	2	pwsgrp 13908	. . . . 5 |- ( ( R e. Grp /\ I e. V ) -> Y e. Grp )
28	4, 18	grpinvcl 13845	. . . . 5 |- ( ( Y e. Grp /\ G e. B ) -> ( ( invg ` Y ) ` G ) e. B )
29	27, 13, 28	syl2an2r 599	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` Y ) ` G ) e. B )
30		eqid 2234	. . . 4 |- ( +g ` R ) = ( +g ` R )
31		eqid 2234	. . . 4 |- ( +g ` Y ) = ( +g ` Y )
32	2, 4, 5, 1, 6, 29, 30, 31	pwsplusgval 13592	. . 3 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F ( +g ` Y ) ( ( invg ` Y ) ` G ) ) = ( F oF ( +g ` R ) ( ( invg ` Y ) ` G ) ) )
33		pwssub.m	. . . . . 6 |- M = ( -g ` R )
34	3, 30, 9, 33	grpsubval 13843	. . . . 5 |- ( ( ( F ` x ) e. ( Base ` R ) /\ ( G ` x ) e. ( Base ` R ) ) -> ( ( F ` x ) M ( G ` x ) ) = ( ( F ` x ) ( +g ` R ) ( ( invg ` R ) ` ( G ` x ) ) ) )
35	8, 15, 34	syl2anc 411	. . . 4 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( ( F ` x ) M ( G ` x ) ) = ( ( F ` x ) ( +g ` R ) ( ( invg ` R ) ` ( G ` x ) ) ) )
36	35	mpteq2dva 4205	. . 3 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( x e. I |-> ( ( F ` x ) M ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( +g ` R ) ( ( invg ` R ) ` ( G ` x ) ) ) ) )
37	26, 32, 36	3eqtr4d 2277	. 2 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F ( +g ` Y ) ( ( invg ` Y ) ` G ) ) = ( x e. I |-> ( ( F ` x ) M ( G ` x ) ) ) )
38		pwssub.n	. . . 4 |- .- = ( -g ` Y )
39	4, 31, 18, 38	grpsubval 13843	. . 3 |- ( ( F e. B /\ G e. B ) -> ( F .- G ) = ( F ( +g ` Y ) ( ( invg ` Y ) ` G ) ) )
40	39	adantl 277	. 2 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F .- G ) = ( F ( +g ` Y ) ( ( invg ` Y ) ` G ) ) )
41	1, 8, 15, 17, 21	offval2 6291	. 2 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F oF M G ) = ( x e. I |-> ( ( F ` x ) M ( G ` x ) ) ) )
42	37, 40, 41	3eqtr4d 2277	1 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F .- G ) = ( F oF M G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   |-> cmpt 4176   o. ccom 4758   -->wf 5353   ` cfv 5357  (class class class)co 6058   oFcof 6273   Basecbs 13296   +g cplusg 13374   ^_s cpws 13563   Grpcgrp 13797   invgcminusg 13798   -gcsg 13799

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-of 6275  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  df-sbg 13802

This theorem is referenced by: (None)