Theorem imasgim 37670

Description: A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.)

Hypotheses

Ref	Expression
imasgim.u	|- ( ph -> U = ( F "s R ) )
imasgim.v	|- ( ph -> V = ( Base ` R ) )
imasgim.f	|- ( ph -> F : V -1-1-onto-> B )
imasgim.r	|- ( ph -> R e. Grp )

Assertion

Ref	Expression
imasgim	|- ( ph -> F e. ( R GrpIso U ) )

Proof of Theorem imasgim

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
2		eqid 2622	. . 3 |- ( Base ` U ) = ( Base ` U )
3		eqid 2622	. . 3 |- ( +g ` R ) = ( +g ` R )
4		eqid 2622	. . 3 |- ( +g ` U ) = ( +g ` U )
5		imasgim.r	. . 3 |- ( ph -> R e. Grp )
6		imasgim.u	. . . . 5 |- ( ph -> U = ( F "s R ) )
7		imasgim.v	. . . . 5 |- ( ph -> V = ( Base ` R ) )
8		eqidd 2623	. . . . 5 |- ( ph -> ( +g ` R ) = ( +g ` R ) )
9		imasgim.f	. . . . . 6 |- ( ph -> F : V -1-1-onto-> B )
10		f1ofo 6144	. . . . . 6 |- ( F : V -1-1-onto-> B -> F : V -onto-> B )
11	9, 10	syl 17	. . . . 5 |- ( ph -> F : V -onto-> B )
12	9	f1ocpbl 16185	. . . . 5 |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( c e. V /\ d e. V ) ) -> ( ( ( F ` a ) = ( F ` c ) /\ ( F ` b ) = ( F ` d ) ) -> ( F ` ( a ( +g ` R ) b ) ) = ( F ` ( c ( +g ` R ) d ) ) ) )
13		eqid 2622	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
14	6, 7, 8, 11, 12, 5, 13	imasgrp 17531	. . . 4 |- ( ph -> ( U e. Grp /\ ( F ` ( 0g ` R ) ) = ( 0g ` U ) ) )
15	14	simpld 475	. . 3 |- ( ph -> U e. Grp )
16	6, 7, 11, 5	imasbas 16172	. . . . . . 7 |- ( ph -> B = ( Base ` U ) )
17		f1oeq3 6129	. . . . . . 7 |- ( B = ( Base ` U ) -> ( F : V -1-1-onto-> B <-> F : V -1-1-onto-> ( Base ` U ) ) )
18	16, 17	syl 17	. . . . . 6 |- ( ph -> ( F : V -1-1-onto-> B <-> F : V -1-1-onto-> ( Base ` U ) ) )
19	9, 18	mpbid 222	. . . . 5 |- ( ph -> F : V -1-1-onto-> ( Base ` U ) )
20		f1oeq2 6128	. . . . . 6 |- ( V = ( Base ` R ) -> ( F : V -1-1-onto-> ( Base ` U ) <-> F : ( Base ` R ) -1-1-onto-> ( Base ` U ) ) )
21	7, 20	syl 17	. . . . 5 |- ( ph -> ( F : V -1-1-onto-> ( Base ` U ) <-> F : ( Base ` R ) -1-1-onto-> ( Base ` U ) ) )
22	19, 21	mpbid 222	. . . 4 |- ( ph -> F : ( Base ` R ) -1-1-onto-> ( Base ` U ) )
23		f1of 6137	. . . 4 |- ( F : ( Base ` R ) -1-1-onto-> ( Base ` U ) -> F : ( Base ` R ) --> ( Base ` U ) )
24	22, 23	syl 17	. . 3 |- ( ph -> F : ( Base ` R ) --> ( Base ` U ) )
25	7	eleq2d 2687	. . . . . 6 |- ( ph -> ( a e. V <-> a e. ( Base ` R ) ) )
26	7	eleq2d 2687	. . . . . 6 |- ( ph -> ( b e. V <-> b e. ( Base ` R ) ) )
27	25, 26	anbi12d 747	. . . . 5 |- ( ph -> ( ( a e. V /\ b e. V ) <-> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) ) ) )
28	11, 12, 6, 7, 5, 3, 4	imasaddval 16192	. . . . . . 7 |- ( ( ph /\ a e. V /\ b e. V ) -> ( ( F ` a ) ( +g ` U ) ( F ` b ) ) = ( F ` ( a ( +g ` R ) b ) ) )
29	28	eqcomd 2628	. . . . . 6 |- ( ( ph /\ a e. V /\ b e. V ) -> ( F ` ( a ( +g ` R ) b ) ) = ( ( F ` a ) ( +g ` U ) ( F ` b ) ) )
30	29	3expib 1268	. . . . 5 |- ( ph -> ( ( a e. V /\ b e. V ) -> ( F ` ( a ( +g ` R ) b ) ) = ( ( F ` a ) ( +g ` U ) ( F ` b ) ) ) )
31	27, 30	sylbird 250	. . . 4 |- ( ph -> ( ( a e. ( Base ` R ) /\ b e. ( Base ` R ) ) -> ( F ` ( a ( +g ` R ) b ) ) = ( ( F ` a ) ( +g ` U ) ( F ` b ) ) ) )
32	31	imp 445	. . 3 |- ( ( ph /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) ) ) -> ( F ` ( a ( +g ` R ) b ) ) = ( ( F ` a ) ( +g ` U ) ( F ` b ) ) )
33	1, 2, 3, 4, 5, 15, 24, 32	isghmd 17669	. 2 |- ( ph -> F e. ( R GrpHom U ) )
34	1, 2	isgim 17704	. 2 |- ( F e. ( R GrpIso U ) <-> ( F e. ( R GrpHom U ) /\ F : ( Base ` R ) -1-1-onto-> ( Base ` U ) ) )
35	33, 22, 34	sylanbrc 698	1 |- ( ph -> F e. ( R GrpIso U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   "_s cimas 16164   Grpcgrp 17422   GrpHom cghm 17657   GrpIso cgim 17699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-imas 16168  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ghm 17658  df-gim 17701

This theorem is referenced by:  isnumbasgrplem1  37671