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Theorem imasgim 27264
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.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2283 . . 3  |-  ( Base `  U )  =  (
Base `  U )
3 eqid 2283 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2283 . . 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 2284 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
9 imasgim.f . . . . . 6  |-  ( ph  ->  F : V -1-1-onto-> B )
10 f1ofo 5479 . . . . . 6  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
119, 10syl 15 . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
129f1ocpbl 13427 . . . . 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 2283 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
146, 7, 8, 11, 12, 5, 13imasgrp 14611 . . . 4  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  ( 0g `  R ) )  =  ( 0g `  U ) ) )
1514simpld 445 . . 3  |-  ( ph  ->  U  e.  Grp )
166, 7, 11, 5imasbas 13415 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  U ) )
17 f1oeq3 5465 . . . . . . 7  |-  ( B  =  ( Base `  U
)  ->  ( F : V -1-1-onto-> B  <->  F : V -1-1-onto-> ( Base `  U ) ) )
1816, 17syl 15 . . . . . 6  |-  ( ph  ->  ( F : V -1-1-onto-> B  <->  F : V -1-1-onto-> ( Base `  U
) ) )
199, 18mpbid 201 . . . . 5  |-  ( ph  ->  F : V -1-1-onto-> ( Base `  U ) )
20 f1oeq2 5464 . . . . . 6  |-  ( V  =  ( Base `  R
)  ->  ( F : V -1-1-onto-> ( Base `  U
)  <->  F : ( Base `  R ) -1-1-onto-> ( Base `  U
) ) )
217, 20syl 15 . . . . 5  |-  ( ph  ->  ( F : V -1-1-onto-> ( Base `  U )  <->  F :
( Base `  R ) -1-1-onto-> ( Base `  U ) ) )
2219, 21mpbid 201 . . . 4  |-  ( ph  ->  F : ( Base `  R ) -1-1-onto-> ( Base `  U
) )
23 f1of 5472 . . . 4  |-  ( F : ( Base `  R
)
-1-1-onto-> ( Base `  U )  ->  F : ( Base `  R ) --> ( Base `  U ) )
2422, 23syl 15 . . 3  |-  ( ph  ->  F : ( Base `  R ) --> ( Base `  U ) )
257eleq2d 2350 . . . . . 6  |-  ( ph  ->  ( a  e.  V  <->  a  e.  ( Base `  R
) ) )
267eleq2d 2350 . . . . . 6  |-  ( ph  ->  ( b  e.  V  <->  b  e.  ( Base `  R
) ) )
2725, 26anbi12d 691 . . . . 5  |-  ( ph  ->  ( ( a  e.  V  /\  b  e.  V )  <->  ( a  e.  ( Base `  R
)  /\  b  e.  ( Base `  R )
) ) )
2811, 12, 6, 7, 5, 3, 4imasaddval 13434 . . . . . . 7  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( ( F `  a )
( +g  `  U ) ( F `  b
) )  =  ( F `  ( a ( +g  `  R
) b ) ) )
2928eqcomd 2288 . . . . . 6  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( F `  ( a ( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U
) ( F `  b ) ) )
30293expib 1154 . . . . 5  |-  ( ph  ->  ( ( a  e.  V  /\  b  e.  V )  ->  ( F `  ( a
( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U ) ( F `
 b ) ) ) )
3127, 30sylbird 226 . . . 4  |-  ( ph  ->  ( ( a  e.  ( Base `  R
)  /\  b  e.  ( Base `  R )
)  ->  ( F `  ( a ( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U
) ( F `  b ) ) ) )
3231imp 418 . . 3  |-  ( (
ph  /\  ( a  e.  ( Base `  R
)  /\  b  e.  ( Base `  R )
) )  ->  ( F `  ( a
( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U ) ( F `
 b ) ) )
331, 2, 3, 4, 5, 15, 24, 32isghmd 14692 . 2  |-  ( ph  ->  F  e.  ( R 
GrpHom  U ) )
341, 2isgim 14726 . 2  |-  ( F  e.  ( R GrpIso  U
)  <->  ( F  e.  ( R  GrpHom  U )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  U ) ) )
3533, 22, 34sylanbrc 645 1  |-  ( ph  ->  F  e.  ( R GrpIso  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400    "s cimas 13407   Grpcgrp 14362    GrpHom cghm 14680   GrpIso cgim 14721
This theorem is referenced by:  isnumbasgrplem1  27266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-mnd 14367  df-grp 14489  df-minusg 14490  df-ghm 14681  df-gim 14723
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