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Theorem imasgim 27140
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 2412 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2412 . . 3  |-  ( Base `  U )  =  (
Base `  U )
3 eqid 2412 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2412 . . 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 2413 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
9 imasgim.f . . . . . 6  |-  ( ph  ->  F : V -1-1-onto-> B )
10 f1ofo 5648 . . . . . 6  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
119, 10syl 16 . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
129f1ocpbl 13713 . . . . 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 2412 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
146, 7, 8, 11, 12, 5, 13imasgrp 14897 . . . 4  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  ( 0g `  R ) )  =  ( 0g `  U ) ) )
1514simpld 446 . . 3  |-  ( ph  ->  U  e.  Grp )
166, 7, 11, 5imasbas 13701 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  U ) )
17 f1oeq3 5634 . . . . . . 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 202 . . . . 5  |-  ( ph  ->  F : V -1-1-onto-> ( Base `  U ) )
20 f1oeq2 5633 . . . . . 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 202 . . . 4  |-  ( ph  ->  F : ( Base `  R ) -1-1-onto-> ( Base `  U
) )
23 f1of 5641 . . . 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 2479 . . . . . 6  |-  ( ph  ->  ( a  e.  V  <->  a  e.  ( Base `  R
) ) )
267eleq2d 2479 . . . . . 6  |-  ( ph  ->  ( b  e.  V  <->  b  e.  ( Base `  R
) ) )
2725, 26anbi12d 692 . . . . 5  |-  ( ph  ->  ( ( a  e.  V  /\  b  e.  V )  <->  ( a  e.  ( Base `  R
)  /\  b  e.  ( Base `  R )
) ) )
2811, 12, 6, 7, 5, 3, 4imasaddval 13720 . . . . . . 7  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( ( F `  a )
( +g  `  U ) ( F `  b
) )  =  ( F `  ( a ( +g  `  R
) b ) ) )
2928eqcomd 2417 . . . . . 6  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( F `  ( a ( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U
) ( F `  b ) ) )
30293expib 1156 . . . . 5  |-  ( ph  ->  ( ( a  e.  V  /\  b  e.  V )  ->  ( F `  ( a
( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U ) ( F `
 b ) ) ) )
3127, 30sylbird 227 . . . 4  |-  ( ph  ->  ( ( a  e.  ( Base `  R
)  /\  b  e.  ( Base `  R )
)  ->  ( F `  ( a ( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U
) ( F `  b ) ) ) )
3231imp 419 . . 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 14978 . 2  |-  ( ph  ->  F  e.  ( R 
GrpHom  U ) )
341, 2isgim 15012 . 2  |-  ( F  e.  ( R GrpIso  U
)  <->  ( F  e.  ( R  GrpHom  U )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  U ) ) )
3533, 22, 34sylanbrc 646 1  |-  ( ph  ->  F  e.  ( R GrpIso  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   -->wf 5417   -onto->wfo 5419   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492   0gc0g 13686    "s cimas 13693   Grpcgrp 14648    GrpHom cghm 14966   GrpIso cgim 15007
This theorem is referenced by:  isnumbasgrplem1  27142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-0g 13690  df-imas 13697  df-mnd 14653  df-grp 14775  df-minusg 14776  df-ghm 14967  df-gim 15009
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