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Theorem frgpup3 17506
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 2471 . . 3  |-  ( invg `  H )  =  ( invg `  H )
3 eqid 2471 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( ( invg `  H ) `
 ( F `  y ) ) ) )  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( ( invg `  H ) `
 ( F `  y ) ) ) )
4 simp1 1030 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  H  e.  Grp )
5 simp2 1031 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  I  e.  V
)
6 simp3 1032 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F : I --> B )
7 eqid 2471 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
8 eqid 2471 . . 3  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
9 frgpup3.g . . 3  |-  G  =  (freeGrp `  I )
10 eqid 2471 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2471 . . 3  |-  ran  (
g  e.  (  _I 
` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( invg `  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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11frgpup1 17503 . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  e.  ( G 
GrpHom  H ) )
134adantr 472 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  H  e.  Grp )
145adantr 472 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  I  e.  V )
156adantr 472 . . . . 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 468 . . . . 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 17504 . . . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) )  =  ( F `  k
) )
1918mpteq2dva 4482 . . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) ) )  =  ( k  e.  I  |->  ( F `  k ) ) )
2010, 1ghmf 16965 . . . . 5  |-  ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( invg `  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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) : ( Base `  G ) --> B )
2112, 20syl 17 . . . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) : ( Base `  G ) --> B )
228, 16, 9, 10vrgpf 17496 . . . . 5  |-  ( I  e.  V  ->  U : I --> ( Base `  G ) )
235, 22syl 17 . . . 4  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  U : I --> ( Base `  G
) )
24 fcompt 6075 . . . 4  |-  ( ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( invg `  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 ) ,  ( ( invg `  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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) ) ) )
2521, 23, 24syl2anc 673 . . 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 ) ,  ( ( invg `  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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) `  ( U `  k ) ) ) )
266feqmptd 5932 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F  =  ( k  e.  I  |->  ( F `  k ) ) )
2719, 25, 263eqtr4d 2515 . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U )  =  F )
284adantr 472 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  H  e.  Grp )
295adantr 472 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  I  e.  V )
306adantr 472 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  F : I --> B )
31 simprl 772 . . . . 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 774 . . . . 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 17505 . . . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) )
3433expr 626 . . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) ) )
3534ralrimiva 2809 . 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) ) )
36 coeq1 4997 . . . 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 ) ,  ( ( invg `  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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U ) )
3736eqeq1d 2473 . . 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 ) ,  ( ( invg `  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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
)  o.  U )  =  F ) )
3837eqreu 3218 . 2  |-  ( ( ran  ( g  e.  (  _I  ` Word  ( I  X.  2o ) ) 
|->  <. [ g ] ( ~FG  `  I ) ,  ( H  gsumg  ( ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `
 y ) ,  ( ( invg `  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 ) ,  ( ( invg `  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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) ) )  ->  E! m  e.  ( G  GrpHom  H ) ( m  o.  U )  =  F )
3912, 27, 35, 38syl3anc 1292 1  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  E! m  e.  ( G  GrpHom  H ) ( m  o.  U
)  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E!wreu 2758   (/)c0 3722   ifcif 3872   <.cop 3965    |-> cmpt 4454    _I cid 4749    X. cxp 4837   ran crn 4840    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   2oc2o 7194   [cec 7379  Word cword 12703   Basecbs 15199    gsumg cgsu 15417   Grpcgrp 16747   invgcminusg 16748    GrpHom cghm 16958   ~FG cefg 17434  freeGrpcfrgp 17435  varFGrpcvrgp 17436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-reverse 12717  df-s2 13003  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-0g 15418  df-gsum 15419  df-imas 15485  df-qus 15487  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-frmd 16711  df-vrmd 16712  df-grp 16751  df-minusg 16752  df-ghm 16959  df-efg 17437  df-frgp 17438  df-vrgp 17439
This theorem is referenced by:  0frgp  17507  frgpcyg  19221
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