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Theorem frgpup3 15365
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 2404 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
3 eqid 2404 . . 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 957 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  H  e.  Grp )
5 simp2 958 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  I  e.  V
)
6 simp3 959 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F : I --> B )
7 eqid 2404 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
8 eqid 2404 . . 3  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
9 frgpup3.g . . 3  |-  G  =  (freeGrp `  I )
10 eqid 2404 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2404 . . 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 15362 . 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 452 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  H  e.  Grp )
145adantr 452 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  I  e.  V )
156adantr 452 . . . . 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 448 . . . . 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 15363 . . . 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 4255 . . 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 14965 . . . . 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 16 . . . 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 15355 . . . . 5  |-  ( I  e.  V  ->  U : I --> ( Base `  G ) )
235, 22syl 16 . . . 4  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  U : I --> ( Base `  G
) )
24 fcompt 5863 . . . 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 643 . . 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 5738 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F  =  ( k  e.  I  |->  ( F `  k ) ) )
2719, 25, 263eqtr4d 2446 . 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 452 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  H  e.  Grp )
295adantr 452 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  I  e.  V )
306adantr 452 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  F : I --> B )
31 simprl 733 . . . . 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 734 . . . . 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 15364 . . . 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 599 . . 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 2749 . 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 4989 . . . 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 2412 . . 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 3086 . 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 1184 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E!wreu 2668   (/)c0 3588   ifcif 3699   <.cop 3777    e. cmpt 4226    _I cid 4453    X. cxp 4835   ran crn 4838    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   2oc2o 6677   [cec 6862  Word cword 11672   Basecbs 13424    gsumg cgsu 13679   Grpcgrp 14640   inv gcminusg 14641    GrpHom cghm 14958   ~FG cefg 15293  freeGrpcfrgp 15294  varFGrpcvrgp 15295
This theorem is referenced by:  0frgp  15366  frgpcyg  16809
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-reverse 11683  df-s2 11767  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-0g 13682  df-gsum 13683  df-imas 13689  df-divs 13690  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-frmd 14749  df-vrmd 14750  df-grp 14767  df-minusg 14768  df-ghm 14959  df-efg 15296  df-frgp 15297  df-vrgp 15298
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