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Theorem frgpup3 15103
Description: Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
frgpup3.g  |-  G  =  (freeGrp `  I )
frgpup3.b  |-  B  =  ( Base `  H
)
frgpup3.u  |-  U  =  (varFGrp `  I )
Assertion
Ref Expression
frgpup3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  E! m  e.  ( G  GrpHom  H ) ( m  o.  U
)  =  F )
Distinct variable groups:    B, m    m, F    m, G    m, H    m, I    U, m   
m, V

Proof of Theorem frgpup3
Dummy variables  g 
k  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup3.b . . 3  |-  B  =  ( Base `  H
)
2 eqid 2296 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
3 eqid 2296 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( ( inv g `  H ) `
 ( F `  y ) ) ) )  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( ( inv g `  H ) `
 ( F `  y ) ) ) )
4 simp1 955 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  H  e.  Grp )
5 simp2 956 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  I  e.  V
)
6 simp3 957 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F : I --> B )
7 eqid 2296 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
8 eqid 2296 . . 3  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
9 frgpup3.g . . 3  |-  G  =  (freeGrp `  I )
10 eqid 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2296 . . 3  |-  ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  =  ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11frgpup1 15100 . 2  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  e.  ( G 
GrpHom  H ) )
134adantr 451 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  H  e.  Grp )
145adantr 451 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  I  e.  V )
156adantr 451 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  F :
I --> B )
16 frgpup3.u . . . . 5  |-  U  =  (varFGrp `  I )
17 simpr 447 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  k  e.  I )
181, 2, 3, 13, 14, 15, 7, 8, 9, 10, 11, 16, 17frgpup2 15101 . . . 4  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) )  =  ( F `  k
) )
1918mpteq2dva 4122 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  ( k  e.  I  |->  ( ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) ) )  =  ( k  e.  I  |->  ( F `  k ) ) )
2010, 1ghmf 14703 . . . . 5  |-  ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  e.  ( G 
GrpHom  H )  ->  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) : ( Base `  G ) --> B )
2112, 20syl 15 . . . 4  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) : ( Base `  G ) --> B )
228, 16, 9, 10vrgpf 15093 . . . . 5  |-  ( I  e.  V  ->  U : I --> ( Base `  G ) )
235, 22syl 15 . . . 4  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  U : I --> ( Base `  G
) )
24 fcompt 5710 . . . 4  |-  ( ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) : ( Base `  G ) --> B  /\  U : I --> ( Base `  G ) )  -> 
( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U )  =  ( k  e.  I  |->  ( ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) ) ) )
2521, 23, 24syl2anc 642 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  ( ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U )  =  ( k  e.  I  |->  ( ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) ) ) )
266feqmptd 5591 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F  =  ( k  e.  I  |->  ( F `  k ) ) )
2719, 25, 263eqtr4d 2338 . 2  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  ( ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U )  =  F )
284adantr 451 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  H  e.  Grp )
295adantr 451 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  I  e.  V )
306adantr 451 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  F : I --> B )
31 simprl 732 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  m  e.  ( G  GrpHom  H ) )
32 simprr 733 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  (
m  o.  U )  =  F )
331, 2, 3, 28, 29, 30, 7, 8, 9, 10, 11, 16, 31, 32frgpup3lem 15102 . . . 4  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  m  =  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) )
3433expr 598 . . 3  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  m  e.  ( G  GrpHom  H ) )  ->  ( ( m  o.  U )  =  F  ->  m  =  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) ) )
3534ralrimiva 2639 . 2  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  A. m  e.  ( G  GrpHom  H ) ( ( m  o.  U
)  =  F  ->  m  =  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) ) )
36 coeq1 4857 . . . 4  |-  ( m  =  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  ->  ( m  o.  U )  =  ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U ) )
3736eqeq1d 2304 . . 3  |-  ( m  =  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  ->  ( (
m  o.  U )  =  F  <->  ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U )  =  F ) )
3837eqreu 2970 . 2  |-  ( ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  e.  ( G 
GrpHom  H )  /\  ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U )  =  F  /\  A. m  e.  ( G  GrpHom  H ) ( ( m  o.  U )  =  F  ->  m  =  ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( inv g `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) ) )  ->  E! m  e.  ( G  GrpHom  H ) ( m  o.  U )  =  F )
3912, 27, 35, 38syl3anc 1182 1  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  E! m  e.  ( G  GrpHom  H ) ( m  o.  U
)  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E!wreu 2558   (/)c0 3468   ifcif 3578   <.cop 3656    e. cmpt 4093    _I cid 4320    X. cxp 4703   ran crn 4706    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2oc2o 6489   [cec 6674  Word cword 11419   Basecbs 13164    gsumg cgsu 13417   Grpcgrp 14378   inv gcminusg 14379    GrpHom cghm 14696   ~FG cefg 15031  freeGrpcfrgp 15032  varFGrpcvrgp 15033
This theorem is referenced by:  0frgp  15104  frgpcyg  16543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-frmd 14487  df-vrmd 14488  df-grp 14505  df-minusg 14506  df-ghm 14697  df-efg 15034  df-frgp 15035  df-vrgp 15036
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