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Theorem frgpup3 16255
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 2433 . . 3  |-  ( invg `  H )  =  ( invg `  H )
3 eqid 2433 . . 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 981 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  H  e.  Grp )
5 simp2 982 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  I  e.  V
)
6 simp3 983 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F : I --> B )
7 eqid 2433 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
8 eqid 2433 . . 3  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
9 frgpup3.g . . 3  |-  G  =  (freeGrp `  I )
10 eqid 2433 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2433 . . 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 16252 . 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 462 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  H  e.  Grp )
145adantr 462 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  k  e.  I
)  ->  I  e.  V )
156adantr 462 . . . . 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 458 . . . . 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 16253 . . . 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 4366 . . 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 15731 . . . . 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 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 ) ,  ( ( invg `  H ) `  ( F `  y )
) ) )  o.  g ) ) >.
) : ( Base `  G ) --> B )
228, 16, 9, 10vrgpf 16245 . . . . 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 5866 . . . 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 654 . . 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 5732 . . 3  |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  ->  F  =  ( k  e.  I  |->  ( F `  k ) ) )
2719, 25, 263eqtr4d 2475 . 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 462 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  H  e.  Grp )
295adantr 462 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  I  e.  V )
306adantr 462 . . . . 5  |-  ( ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B )  /\  ( m  e.  ( G  GrpHom  H )  /\  ( m  o.  U )  =  F ) )  ->  F : I --> B )
31 simprl 748 . . . . 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 749 . . . . 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 16254 . . . 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 610 . . 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 2789 . 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 4984 . . . 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 2441 . . 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 3140 . 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 1211 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 369    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705   E!wreu 2707   (/)c0 3625   ifcif 3779   <.cop 3871    e. cmpt 4338    _I cid 4618    X. cxp 4825   ran crn 4828    o. ccom 4831   -->wf 5402   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   2oc2o 6902   [cec 7087  Word cword 12205   Basecbs 14157    gsumg cgsu 14362   Grpcgrp 15393   invgcminusg 15394    GrpHom cghm 15724   ~FG cefg 16183  freeGrpcfrgp 16184  varFGrpcvrgp 16185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-ot 3874  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-ec 7091  df-qs 7095  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-fz 11425  df-fzo 11533  df-seq 11791  df-hash 12088  df-word 12213  df-concat 12215  df-s1 12216  df-substr 12217  df-splice 12218  df-reverse 12219  df-s2 12459  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-0g 14363  df-gsum 14364  df-imas 14429  df-divs 14430  df-mnd 15398  df-mhm 15447  df-submnd 15448  df-frmd 15507  df-vrmd 15508  df-grp 15525  df-minusg 15526  df-ghm 15725  df-efg 16186  df-frgp 16187  df-vrgp 16188
This theorem is referenced by:  0frgp  16256  frgpcyg  17848
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