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Theorem frgpupf 15082
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 14494 . . . . . . 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 5580 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
86, 7eqsstri 3208 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
98sseli 3176 . . . . . . 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 15079 . . . . . . 7  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
1716adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  W )  ->  T : ( I  X.  2o ) --> B )
18 wrdco 11486 . . . . . 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 14463 . . . . 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 15027 . . . . 5  |-  .~  Er  W
2423a1i 10 . . . 4  |-  ( ph  ->  .~  Er  W )
25 fvex 5539 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
266, 25eqeltri 2353 . . . . 5  |-  W  e. 
_V
2726a1i 10 . . . 4  |-  ( ph  ->  W  e.  _V )
28 coeq2 4842 . . . . 5  |-  ( g  =  h  ->  ( T  o.  g )  =  ( T  o.  h ) )
2928oveq2d 5874 . . . 4  |-  ( g  =  h  ->  ( H  gsumg  ( T  o.  g
) )  =  ( H  gsumg  ( T  o.  h
) ) )
3011, 12, 13, 2, 14, 15, 6, 22frgpuplem 15081 . . . 4  |-  ( (
ph  /\  g  .~  h )  ->  ( H  gsumg  ( T  o.  g
) )  =  ( H  gsumg  ( T  o.  h
) ) )
311, 21, 24, 27, 29, 30qliftfund 6744 . . 3  |-  ( ph  ->  Fun  E )
321, 21, 24, 27qliftf 6746 . . 3  |-  ( ph  ->  ( Fun  E  <->  E :
( W /.  .~  )
--> B ) )
3331, 32mpbid 201 . 2  |-  ( ph  ->  E : ( W /.  .~  ) --> B )
34 frgpup.g . . . . . . 7  |-  G  =  (freeGrp `  I )
35 eqid 2283 . . . . . . 7  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3634, 35, 22frgpval 15067 . . . . . 6  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
3714, 36syl 15 . . . . 5  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
38 2on 6487 . . . . . . . . 9  |-  2o  e.  On
39 xpexg 4800 . . . . . . . . 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 11425 . . . . . . . 8  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
42 fvi 5579 . . . . . . . 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 2327 . . . . . 6  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
45 eqid 2283 . . . . . . . 8  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
4635, 45frmdbas 14474 . . . . . . 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 2318 . . . . 5  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
49 fvex 5539 . . . . . . 7  |-  ( ~FG  `  I
)  e.  _V
5022, 49eqeltri 2353 . . . . . 6  |-  .~  e.  _V
5150a1i 10 . . . . 5  |-  ( ph  ->  .~  e.  _V )
52 fvex 5539 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5352a1i 10 . . . . 5  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
5437, 48, 51, 53divsbas 13447 . . . 4  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
55 frgpup.x . . . 4  |-  X  =  ( Base `  G
)
5654, 55syl6reqr 2334 . . 3  |-  ( ph  ->  X  =  ( W /.  .~  ) )
5756feq2d 5380 . 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 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ifcif 3565   <.cop 3643    e. cmpt 4077    _I cid 4304   Oncon0 4392    X. cxp 4687   ran crn 4690    o. ccom 4693   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2oc2o 6473    Er wer 6657   [cec 6658   /.cqs 6659  Word cword 11403   Basecbs 13148    gsumg cgsu 13401    /.s cqus 13408   Mndcmnd 14361   Grpcgrp 14362   inv gcminusg 14363  freeMndcfrmd 14469   ~FG cefg 15015  freeGrpcfrgp 15016
This theorem is referenced by:  frgpupval  15083  frgpup1  15084
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-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-submnd 14416  df-frmd 14471  df-grp 14489  df-minusg 14490  df-efg 15018  df-frgp 15019
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