Theorem pwssub 13826

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 2232	. . . . . 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 13507	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F : I --> ( Base ` R ) )
8	7	ffvelcdmda 5812	. . . 4 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( F ` x ) e. ( Base ` R ) )
9		eqid 2232	. . . . . . . 8 |- ( invg ` R ) = ( invg ` R )
10	3, 9	grpinvf 13760	. . . . . . 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 13507	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G : I --> ( Base ` R ) )
15	14	ffvelcdmda 5812	. . . . 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 5813	. . . 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 5730	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F = ( x e. I |-> ( F ` x ) ) )
18		eqid 2232	. . . . . . 7 |- ( invg ` Y ) = ( invg ` Y )
19	2, 4, 9, 18	pwsinvg 13825	. . . . . 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 5730	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G = ( x e. I |-> ( G ` x ) ) )
22	11	feqmptd 5730	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( invg ` R ) = ( y e. ( Base ` R ) |-> ( ( invg ` R ) ` y ) ) )
23		fveq2 5670	. . . . . 6 |- ( y = ( G ` x ) -> ( ( invg ` R ) ` y ) = ( ( invg ` R ) ` ( G ` x ) ) )
24	15, 21, 22, 23	fmptco 5843	. . . . 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 2265	. . . 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 6282	. . 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 13824	. . . . 5 |- ( ( R e. Grp /\ I e. V ) -> Y e. Grp )
28	4, 18	grpinvcl 13761	. . . . 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 2232	. . . 4 |- ( +g ` R ) = ( +g ` R )
31		eqid 2232	. . . 4 |- ( +g ` Y ) = ( +g ` Y )
32	2, 4, 5, 1, 6, 29, 30, 31	pwsplusgval 13508	. . 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 13759	. . . . 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 4200	. . 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 2275	. 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 13759	. . 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 6282	. 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 2275	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 2203   |-> cmpt 4171   o. ccom 4753   -->wf 5348   ` cfv 5352  (class class class)co 6050   oFcof 6264   Basecbs 13212   +g cplusg 13290   ^_s cpws 13479   Grpcgrp 13713   invgcminusg 13714   -gcsg 13715

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 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-of 6266  df-1st 6334  df-2nd 6335  df-map 6884  df-ixp 6934  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-dec 9710  df-uz 9854  df-fz 10343  df-struct 13214  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mulr 13304  df-sca 13306  df-vsca 13307  df-ip 13308  df-tset 13309  df-ple 13310  df-ds 13312  df-hom 13314  df-cco 13315  df-rest 13454  df-topn 13455  df-0g 13471  df-topgen 13473  df-pt 13474  df-prds 13480  df-pws 13503  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-sbg 13718

This theorem is referenced by: (None)