Theorem pwssub 14199

Description: Subtraction in a structure 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 533	. . . 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 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4		pwsinvg.b	. . . . . 6 |- B = ( Base ` Y )
5		simpll 531	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> R e. Grp )
6		simprl 535	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F e. B )
7	2, 3, 4, 5, 1, 6	pwselbas 14190	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F : I --> ( Base ` R ) )
8	7	ffvelcdmda 5837	. . . 4 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( F ` x ) e. ( Base ` R ) )
9		eqid 2238	. . . . . . . 8 |- ( invg ` R ) = ( invg ` R )
10	3, 9	grpinvf 13835	. . . . . . 7 |- ( R e. Grp -> ( invg ` R ) : ( Base ` R ) --> ( Base ` R ) )
11	10	ad2antrr 492	. . . . . 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 537	. . . . . . 7 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G e. B )
14	2, 3, 4, 5, 1, 13	pwselbas 14190	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G : I --> ( Base ` R ) )
15	14	ffvelcdmda 5837	. . . . 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 5838	. . . 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 5753	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F = ( x e. I |-> ( F ` x ) ) )
18		eqid 2238	. . . . . . 7 |- ( invg ` Y ) = ( invg ` Y )
19	2, 4, 9, 18	pwsinvg 14198	. . . . . 6 |- ( ( R e. Grp /\ I e. V /\ G e. B ) -> ( ( invg ` Y ) ` G ) = ( ( invg ` R ) o. G ) )
20	5, 1, 13, 19	syl3anc 1278	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` Y ) ` G ) = ( ( invg ` R ) o. G ) )
21	14	feqmptd 5753	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G = ( x e. I |-> ( G ` x ) ) )
22	11	feqmptd 5753	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( invg ` R ) = ( y e. ( Base ` R ) |-> ( ( invg ` R ) ` y ) ) )
23		fveq2 5693	. . . . . 6 |- ( y = ( G ` x ) -> ( ( invg ` R ) ` y ) = ( ( invg ` R ) ` ( G ` x ) ) )
24	15, 21, 22, 23	fmptco 5868	. . . . 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 2271	. . . 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 6312	. . 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 14197	. . . . 5 |- ( ( R e. Grp /\ I e. V ) -> Y e. Grp )
28	4, 18	grpinvcl 13836	. . . . 5 |- ( ( Y e. Grp /\ G e. B ) -> ( ( invg ` Y ) ` G ) e. B )
29	27, 13, 28	syl2an2r 603	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` Y ) ` G ) e. B )
30		eqid 2238	. . . 4 |- ( +g ` R ) = ( +g ` R )
31		eqid 2238	. . . 4 |- ( +g ` Y ) = ( +g ` Y )
32	2, 4, 5, 1, 6, 29, 30, 31	pwsplusgval 14191	. . 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 13834	. . . . 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 415	. . . 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 4219	. . 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 2281	. 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 13834	. . 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 6312	. 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 2281	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 1402   e. wcel 2209   |-> cmpt 4190   o. ccom 4776   -->wf 5371   ` cfv 5375  (class class class)co 6079   oFcof 6294   Basecbs 13335   +g cplusg 13414   Grpcgrp 13788   invgcminusg 13789   -gcsg 13790   ^_s cpws 14185

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-1st 6368  df-2nd 6369  df-map 6918  df-ixp 6975  df-sup 7318  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-fz 10395  df-struct 13337  df-ndx 13338  df-slot 13339  df-base 13341  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-ip 13432  df-tset 13433  df-ple 13434  df-ds 13436  df-hom 13438  df-cco 13439  df-rest 13578  df-topn 13579  df-0g 13595  df-topgen 13597  df-pt 13598  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-prds 14153  df-pws 14186

This theorem is referenced by: (None)