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Theorem imasgim 29602
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 2454 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2454 . . 3  |-  ( Base `  U )  =  (
Base `  U )
3 eqid 2454 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2454 . . 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 2455 . . . . 5  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
9 imasgim.f . . . . . 6  |-  ( ph  ->  F : V -1-1-onto-> B )
10 f1ofo 5755 . . . . . 6  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
119, 10syl 16 . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
129f1ocpbl 14581 . . . . 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 2454 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
146, 7, 8, 11, 12, 5, 13imasgrp 15789 . . . 4  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  ( 0g `  R ) )  =  ( 0g `  U ) ) )
1514simpld 459 . . 3  |-  ( ph  ->  U  e.  Grp )
166, 7, 11, 5imasbas 14568 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  U ) )
17 f1oeq3 5741 . . . . . . 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 5740 . . . . . 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 5748 . . . 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 2524 . . . . . 6  |-  ( ph  ->  ( a  e.  V  <->  a  e.  ( Base `  R
) ) )
267eleq2d 2524 . . . . . 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 14588 . . . . . . 7  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( ( F `  a )
( +g  `  U ) ( F `  b
) )  =  ( F `  ( a ( +g  `  R
) b ) ) )
2928eqcomd 2462 . . . . . 6  |-  ( (
ph  /\  a  e.  V  /\  b  e.  V
)  ->  ( F `  ( a ( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  U
) ( F `  b ) ) )
30293expib 1191 . . . . 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 15874 . 2  |-  ( ph  ->  F  e.  ( R 
GrpHom  U ) )
341, 2isgim 15908 . 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 1370    e. wcel 1758   -->wf 5521   -onto->wfo 5523   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199   Basecbs 14291   +g cplusg 14356   0gc0g 14496    "s cimas 14560   Grpcgrp 15528    GrpHom cghm 15862   GrpIso cgim 15903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-fz 11554  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-plusg 14369  df-mulr 14370  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-0g 14498  df-imas 14564  df-mnd 15533  df-grp 15663  df-minusg 15664  df-ghm 15863  df-gim 15905
This theorem is referenced by:  isnumbasgrplem1  29604
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