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Theorem frgpupf 15098
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpup.b  |-  B  =  ( Base `  H
)
frgpup.n  |-  N  =  ( inv g `  H )
frgpup.t  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
frgpup.h  |-  ( ph  ->  H  e.  Grp )
frgpup.i  |-  ( ph  ->  I  e.  V )
frgpup.a  |-  ( ph  ->  F : I --> B )
frgpup.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpup.r  |-  .~  =  ( ~FG  `  I )
frgpup.g  |-  G  =  (freeGrp `  I )
frgpup.x  |-  X  =  ( Base `  G
)
frgpup.e  |-  E  =  ran  ( g  e.  W  |->  <. [ g ]  .~  ,  ( H 
gsumg  ( T  o.  g
) ) >. )
Assertion
Ref Expression
frgpupf  |-  ( ph  ->  E : X --> B )
Distinct variable groups:    y, g,
z    g, H    y, F, z    y, N, z    B, g, y, z    T, g    .~ , g    ph, g, y, z    y, I, z   
g, W
Allowed substitution hints:    .~ ( y, z)    T( y, z)    E( y, z, g)    F( g)    G( y, z, g)    H( y, z)    I( g)    N( g)    V( y, z, g)    W( y, z)    X( y, z, g)

Proof of Theorem frgpupf
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 frgpup.e . . . 4  |-  E  =  ran  ( g  e.  W  |->  <. [ g ]  .~  ,  ( H 
gsumg  ( T  o.  g
) ) >. )
2 frgpup.h . . . . . . 7  |-  ( ph  ->  H  e.  Grp )
3 grpmnd 14510 . . . . . . 7  |-  ( H  e.  Grp  ->  H  e.  Mnd )
42, 3syl 15 . . . . . 6  |-  ( ph  ->  H  e.  Mnd )
54adantr 451 . . . . 5  |-  ( (
ph  /\  g  e.  W )  ->  H  e.  Mnd )
6 frgpup.w . . . . . . . . 9  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
7 fviss 5596 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
86, 7eqsstri 3221 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
98sseli 3189 . . . . . . 7  |-  ( g  e.  W  ->  g  e. Word  ( I  X.  2o ) )
109adantl 452 . . . . . 6  |-  ( (
ph  /\  g  e.  W )  ->  g  e. Word  ( I  X.  2o ) )
11 frgpup.b . . . . . . . 8  |-  B  =  ( Base `  H
)
12 frgpup.n . . . . . . . 8  |-  N  =  ( inv g `  H )
13 frgpup.t . . . . . . . 8  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
14 frgpup.i . . . . . . . 8  |-  ( ph  ->  I  e.  V )
15 frgpup.a . . . . . . . 8  |-  ( ph  ->  F : I --> B )
1611, 12, 13, 2, 14, 15frgpuptf 15095 . . . . . . 7  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
1716adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  W )  ->  T : ( I  X.  2o ) --> B )
18 wrdco 11502 . . . . . 6  |-  ( ( g  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  g
)  e. Word  B )
1910, 17, 18syl2anc 642 . . . . 5  |-  ( (
ph  /\  g  e.  W )  ->  ( T  o.  g )  e. Word  B )
2011gsumwcl 14479 . . . . 5  |-  ( ( H  e.  Mnd  /\  ( T  o.  g
)  e. Word  B )  ->  ( H  gsumg  ( T  o.  g
) )  e.  B
)
215, 19, 20syl2anc 642 . . . 4  |-  ( (
ph  /\  g  e.  W )  ->  ( H  gsumg  ( T  o.  g
) )  e.  B
)
22 frgpup.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
236, 22efger 15043 . . . . 5  |-  .~  Er  W
2423a1i 10 . . . 4  |-  ( ph  ->  .~  Er  W )
25 fvex 5555 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
266, 25eqeltri 2366 . . . . 5  |-  W  e. 
_V
2726a1i 10 . . . 4  |-  ( ph  ->  W  e.  _V )
28 coeq2 4858 . . . . 5  |-  ( g  =  h  ->  ( T  o.  g )  =  ( T  o.  h ) )
2928oveq2d 5890 . . . 4  |-  ( g  =  h  ->  ( H  gsumg  ( T  o.  g
) )  =  ( H  gsumg  ( T  o.  h
) ) )
3011, 12, 13, 2, 14, 15, 6, 22frgpuplem 15097 . . . 4  |-  ( (
ph  /\  g  .~  h )  ->  ( H  gsumg  ( T  o.  g
) )  =  ( H  gsumg  ( T  o.  h
) ) )
311, 21, 24, 27, 29, 30qliftfund 6760 . . 3  |-  ( ph  ->  Fun  E )
321, 21, 24, 27qliftf 6762 . . 3  |-  ( ph  ->  ( Fun  E  <->  E :
( W /.  .~  )
--> B ) )
3331, 32mpbid 201 . 2  |-  ( ph  ->  E : ( W /.  .~  ) --> B )
34 frgpup.g . . . . . . 7  |-  G  =  (freeGrp `  I )
35 eqid 2296 . . . . . . 7  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3634, 35, 22frgpval 15083 . . . . . 6  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
3714, 36syl 15 . . . . 5  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
38 2on 6503 . . . . . . . . 9  |-  2o  e.  On
39 xpexg 4816 . . . . . . . . 9  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
4014, 38, 39sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
41 wrdexg 11441 . . . . . . . 8  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
42 fvi 5595 . . . . . . . 8  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
4340, 41, 423syl 18 . . . . . . 7  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
446, 43syl5eq 2340 . . . . . 6  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
45 eqid 2296 . . . . . . . 8  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
4635, 45frmdbas 14490 . . . . . . 7  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
4740, 46syl 15 . . . . . 6  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
4844, 47eqtr4d 2331 . . . . 5  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
49 fvex 5555 . . . . . . 7  |-  ( ~FG  `  I
)  e.  _V
5022, 49eqeltri 2366 . . . . . 6  |-  .~  e.  _V
5150a1i 10 . . . . 5  |-  ( ph  ->  .~  e.  _V )
52 fvex 5555 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5352a1i 10 . . . . 5  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
5437, 48, 51, 53divsbas 13463 . . . 4  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
55 frgpup.x . . . 4  |-  X  =  ( Base `  G
)
5654, 55syl6reqr 2347 . . 3  |-  ( ph  ->  X  =  ( W /.  .~  ) )
5756feq2d 5396 . 2  |-  ( ph  ->  ( E : X --> B 
<->  E : ( W /.  .~  ) --> B ) )
5833, 57mpbird 223 1  |-  ( ph  ->  E : X --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ifcif 3578   <.cop 3656    e. cmpt 4093    _I cid 4320   Oncon0 4408    X. cxp 4703   ran crn 4706    o. ccom 4709   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2oc2o 6489    Er wer 6673   [cec 6674   /.cqs 6675  Word cword 11419   Basecbs 13164    gsumg cgsu 13417    /.s cqus 13424   Mndcmnd 14377   Grpcgrp 14378   inv gcminusg 14379  freeMndcfrmd 14485   ~FG cefg 15031  freeGrpcfrgp 15032
This theorem is referenced by:  frgpupval  15099  frgpup1  15100
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-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-submnd 14432  df-frmd 14487  df-grp 14505  df-minusg 14506  df-efg 15034  df-frgp 15035
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