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Theorem frgpup3 15087
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 2283 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
3 eqid 2283 . . 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 2283 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
8 eqid 2283 . . 3  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
9 frgpup3.g . . 3  |-  G  =  (freeGrp `  I )
10 eqid 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2283 . . 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 15084 . 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 15085 . . . 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 4106 . . 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 14687 . . . . 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 15077 . . . . 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 5694 . . . 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 5575 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F  =  ( k  e.  I  |->  ( F `  k ) ) )
2719, 25, 263eqtr4d 2325 . 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 15086 . . . 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 2626 . 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 4841 . . . 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 2291 . . 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 2957 . 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 1623    e. wcel 1684   A.wral 2543   E!wreu 2545   (/)c0 3455   ifcif 3565   <.cop 3643    e. cmpt 4077    _I cid 4304    X. cxp 4687   ran crn 4690    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2oc2o 6473   [cec 6658  Word cword 11403   Basecbs 13148    gsumg cgsu 13401   Grpcgrp 14362   inv gcminusg 14363    GrpHom cghm 14680   ~FG cefg 15015  freeGrpcfrgp 15016  varFGrpcvrgp 15017
This theorem is referenced by:  0frgp  15088  frgpcyg  16527
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-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  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-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  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-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  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-gsum 13405  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-frmd 14471  df-vrmd 14472  df-grp 14489  df-minusg 14490  df-ghm 14681  df-efg 15018  df-frgp 15019  df-vrgp 15020
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