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Theorem imasgim 29408
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 2438 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2438 . . 3  |-  ( Base `  U )  =  (
Base `  U )
3 eqid 2438 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2438 . . 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 2439 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
9 imasgim.f . . . . . 6  |-  ( ph  ->  F : V -1-1-onto-> B )
10 f1ofo 5643 . . . . . 6  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
119, 10syl 16 . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
129f1ocpbl 14455 . . . . 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 2438 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
146, 7, 8, 11, 12, 5, 13imasgrp 15662 . . . 4  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  ( 0g `  R ) )  =  ( 0g `  U ) ) )
1514simpld 459 . . 3  |-  ( ph  ->  U  e.  Grp )
166, 7, 11, 5imasbas 14442 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  U ) )
17 f1oeq3 5629 . . . . . . 7  |-  ( B  =  ( Base `  U
)  ->  ( F : V -1-1-onto-> B  <->  F : V -1-1-onto-> ( Base `  U ) ) )
1816, 17syl 16 . . . . . 6  |-  ( ph  ->  ( F : V -1-1-onto-> B  <->  F : V -1-1-onto-> ( Base `  U
) ) )
199, 18mpbid 210 . . . . 5  |-  ( ph  ->  F : V -1-1-onto-> ( Base `  U ) )
20 f1oeq2 5628 . . . . . 6  |-  ( V  =  ( Base `  R
)  ->  ( F : V -1-1-onto-> ( Base `  U
)  <->  F : ( Base `  R ) -1-1-onto-> ( Base `  U
) ) )
217, 20syl 16 . . . . 5  |-  ( ph  ->  ( F : V -1-1-onto-> ( Base `  U )  <->  F :
( Base `  R ) -1-1-onto-> ( Base `  U ) ) )
2219, 21mpbid 210 . . . 4  |-  ( ph  ->  F : ( Base `  R ) -1-1-onto-> ( Base `  U
) )
23 f1of 5636 . . . 4  |-  ( F : ( Base `  R
)
-1-1-onto-> ( Base `  U )  ->  F : ( Base `  R ) --> ( Base `  U ) )
2422, 23syl 16 . . 3  |-  ( ph  ->  F : ( Base `  R ) --> ( Base `  U ) )
257eleq2d 2505 . . . . . 6  |-  ( ph  ->  ( a  e.  V  <->  a  e.  ( Base `  R
) ) )
267eleq2d 2505 . . . . . 6  |-  ( ph  ->  ( b  e.  V  <->  b  e.  ( Base `  R
) ) )
2725, 26anbi12d 710 . . . . 5  |-  ( ph  ->  ( ( a  e.  V  /\  b  e.  V )  <->  ( a  e.  ( Base `  R
)  /\  b  e.  ( Base `  R )
) ) )
2811, 12, 6, 7, 5, 3, 4imasaddval 14462 . . . . . . 7  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( ( F `  a )
( +g  `  U ) ( F `  b
) )  =  ( F `  ( a ( +g  `  R
) b ) ) )
2928eqcomd 2443 . . . . . 6  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( F `  ( a ( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U
) ( F `  b ) ) )
30293expib 1190 . . . . 5  |-  ( ph  ->  ( ( a  e.  V  /\  b  e.  V )  ->  ( F `  ( a
( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U ) ( F `
 b ) ) ) )
3127, 30sylbird 235 . . . 4  |-  ( ph  ->  ( ( a  e.  ( Base `  R
)  /\  b  e.  ( Base `  R )
)  ->  ( F `  ( a ( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U
) ( F `  b ) ) ) )
3231imp 429 . . 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 15747 . 2  |-  ( ph  ->  F  e.  ( R 
GrpHom  U ) )
341, 2isgim 15781 . 2  |-  ( F  e.  ( R GrpIso  U
)  <->  ( F  e.  ( R  GrpHom  U )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  U ) ) )
3533, 22, 34sylanbrc 664 1  |-  ( ph  ->  F  e.  ( R GrpIso  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   -->wf 5409   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   0gc0g 14370    "s cimas 14434   Grpcgrp 15402    GrpHom cghm 15735   GrpIso cgim 15776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-0g 14372  df-imas 14438  df-mnd 15407  df-grp 15536  df-minusg 15537  df-ghm 15736  df-gim 15778
This theorem is referenced by:  isnumbasgrplem1  29410
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