Theorem pwssub 13560

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 528	. . . 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 2207	. . . . . 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 529	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F e. B )
7	2, 3, 4, 5, 1, 6	pwselbas 13241	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F : I --> ( Base ` R ) )
8	7	ffvelcdmda 5738	. . . 4 |- ( ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) /\ x e. I ) -> ( F ` x ) e. ( Base ` R ) )
9		eqid 2207	. . . . . . . 8 |- ( invg ` R ) = ( invg ` R )
10	3, 9	grpinvf 13494	. . . . . . 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 531	. . . . . . 7 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G e. B )
14	2, 3, 4, 5, 1, 13	pwselbas 13241	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G : I --> ( Base ` R ) )
15	14	ffvelcdmda 5738	. . . . 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 5739	. . . 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 5655	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> F = ( x e. I |-> ( F ` x ) ) )
18		eqid 2207	. . . . . . 7 |- ( invg ` Y ) = ( invg ` Y )
19	2, 4, 9, 18	pwsinvg 13559	. . . . . 6 |- ( ( R e. Grp /\ I e. V /\ G e. B ) -> ( ( invg ` Y ) ` G ) = ( ( invg ` R ) o. G ) )
20	5, 1, 13, 19	syl3anc 1250	. . . . 5 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` Y ) ` G ) = ( ( invg ` R ) o. G ) )
21	14	feqmptd 5655	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> G = ( x e. I |-> ( G ` x ) ) )
22	11	feqmptd 5655	. . . . . 6 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( invg ` R ) = ( y e. ( Base ` R ) |-> ( ( invg ` R ) ` y ) ) )
23		fveq2 5599	. . . . . 6 |- ( y = ( G ` x ) -> ( ( invg ` R ) ` y ) = ( ( invg ` R ) ` ( G ` x ) ) )
24	15, 21, 22, 23	fmptco 5769	. . . . 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 2240	. . . 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 6197	. . 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 13558	. . . . 5 |- ( ( R e. Grp /\ I e. V ) -> Y e. Grp )
28	4, 18	grpinvcl 13495	. . . . 5 |- ( ( Y e. Grp /\ G e. B ) -> ( ( invg ` Y ) ` G ) e. B )
29	27, 13, 28	syl2an2r 595	. . . 4 |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( ( invg ` Y ) ` G ) e. B )
30		eqid 2207	. . . 4 |- ( +g ` R ) = ( +g ` R )
31		eqid 2207	. . . 4 |- ( +g ` Y ) = ( +g ` Y )
32	2, 4, 5, 1, 6, 29, 30, 31	pwsplusgval 13242	. . 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 13493	. . . . 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 4150	. . 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 2250	. 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 13493	. . 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 6197	. 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 2250	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 1373   e. wcel 2178   |-> cmpt 4121   o. ccom 4697   -->wf 5286   ` cfv 5290  (class class class)co 5967   oFcof 6179   Basecbs 12947   +g cplusg 13024   ^_s cpws 13213   Grpcgrp 13447   invgcminusg 13448   -gcsg 13449

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181  df-1st 6249  df-2nd 6250  df-map 6760  df-ixp 6809  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-n0 9331  df-z 9408  df-dec 9540  df-uz 9684  df-fz 10166  df-struct 12949  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-mulr 13038  df-sca 13040  df-vsca 13041  df-ip 13042  df-tset 13043  df-ple 13044  df-ds 13046  df-hom 13048  df-cco 13049  df-rest 13188  df-topn 13189  df-0g 13205  df-topgen 13207  df-pt 13208  df-prds 13214  df-pws 13237  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-sbg 13452

This theorem is referenced by: (None)