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Theorem frgpupf 15325
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 14737 . . . . . . 7  |-  ( H  e.  Grp  ->  H  e.  Mnd )
42, 3syl 16 . . . . . 6  |-  ( ph  ->  H  e.  Mnd )
54adantr 452 . . . . 5  |-  ( (
ph  /\  g  e.  W )  ->  H  e.  Mnd )
6 frgpup.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
7 fviss 5716 . . . . . . . 8  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
86, 7eqsstri 3314 . . . . . . 7  |-  W  C_ Word  ( I  X.  2o )
98sseli 3280 . . . . . 6  |-  ( g  e.  W  ->  g  e. Word  ( I  X.  2o ) )
10 frgpup.b . . . . . . 7  |-  B  =  ( Base `  H
)
11 frgpup.n . . . . . . 7  |-  N  =  ( inv g `  H )
12 frgpup.t . . . . . . 7  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
13 frgpup.i . . . . . . 7  |-  ( ph  ->  I  e.  V )
14 frgpup.a . . . . . . 7  |-  ( ph  ->  F : I --> B )
1510, 11, 12, 2, 13, 14frgpuptf 15322 . . . . . 6  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
16 wrdco 11720 . . . . . 6  |-  ( ( g  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  g
)  e. Word  B )
179, 15, 16syl2anr 465 . . . . 5  |-  ( (
ph  /\  g  e.  W )  ->  ( T  o.  g )  e. Word  B )
1810gsumwcl 14706 . . . . 5  |-  ( ( H  e.  Mnd  /\  ( T  o.  g
)  e. Word  B )  ->  ( H  gsumg  ( T  o.  g
) )  e.  B
)
195, 17, 18syl2anc 643 . . . 4  |-  ( (
ph  /\  g  e.  W )  ->  ( H  gsumg  ( T  o.  g
) )  e.  B
)
20 frgpup.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
216, 20efger 15270 . . . . 5  |-  .~  Er  W
2221a1i 11 . . . 4  |-  ( ph  ->  .~  Er  W )
23 fvex 5675 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
246, 23eqeltri 2450 . . . . 5  |-  W  e. 
_V
2524a1i 11 . . . 4  |-  ( ph  ->  W  e.  _V )
26 coeq2 4964 . . . . 5  |-  ( g  =  h  ->  ( T  o.  g )  =  ( T  o.  h ) )
2726oveq2d 6029 . . . 4  |-  ( g  =  h  ->  ( H  gsumg  ( T  o.  g
) )  =  ( H  gsumg  ( T  o.  h
) ) )
2810, 11, 12, 2, 13, 14, 6, 20frgpuplem 15324 . . . 4  |-  ( (
ph  /\  g  .~  h )  ->  ( H  gsumg  ( T  o.  g
) )  =  ( H  gsumg  ( T  o.  h
) ) )
291, 19, 22, 25, 27, 28qliftfund 6919 . . 3  |-  ( ph  ->  Fun  E )
301, 19, 22, 25qliftf 6921 . . 3  |-  ( ph  ->  ( Fun  E  <->  E :
( W /.  .~  )
--> B ) )
3129, 30mpbid 202 . 2  |-  ( ph  ->  E : ( W /.  .~  ) --> B )
32 frgpup.g . . . . . . 7  |-  G  =  (freeGrp `  I )
33 eqid 2380 . . . . . . 7  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3432, 33, 20frgpval 15310 . . . . . 6  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
3513, 34syl 16 . . . . 5  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
36 2on 6661 . . . . . . . . 9  |-  2o  e.  On
37 xpexg 4922 . . . . . . . . 9  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
3813, 36, 37sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
39 wrdexg 11659 . . . . . . . 8  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
40 fvi 5715 . . . . . . . 8  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
4138, 39, 403syl 19 . . . . . . 7  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
426, 41syl5eq 2424 . . . . . 6  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
43 eqid 2380 . . . . . . . 8  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
4433, 43frmdbas 14717 . . . . . . 7  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
4538, 44syl 16 . . . . . 6  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
4642, 45eqtr4d 2415 . . . . 5  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
47 fvex 5675 . . . . . . 7  |-  ( ~FG  `  I
)  e.  _V
4820, 47eqeltri 2450 . . . . . 6  |-  .~  e.  _V
4948a1i 11 . . . . 5  |-  ( ph  ->  .~  e.  _V )
50 fvex 5675 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5150a1i 11 . . . . 5  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
5235, 46, 49, 51divsbas 13690 . . . 4  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
53 frgpup.x . . . 4  |-  X  =  ( Base `  G
)
5452, 53syl6reqr 2431 . . 3  |-  ( ph  ->  X  =  ( W /.  .~  ) )
5554feq2d 5514 . 2  |-  ( ph  ->  ( E : X --> B 
<->  E : ( W /.  .~  ) --> B ) )
5631, 55mpbird 224 1  |-  ( ph  ->  E : X --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   (/)c0 3564   ifcif 3675   <.cop 3753    e. cmpt 4200    _I cid 4427   Oncon0 4515    X. cxp 4809   ran crn 4812    o. ccom 4815   Fun wfun 5381   -->wf 5383   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   2oc2o 6647    Er wer 6831   [cec 6832   /.cqs 6833  Word cword 11637   Basecbs 13389    gsumg cgsu 13644    /.s cqus 13651   Mndcmnd 14604   Grpcgrp 14605   inv gcminusg 14606  freeMndcfrmd 14712   ~FG cefg 15258  freeGrpcfrgp 15259
This theorem is referenced by:  frgpupval  15326  frgpup1  15327
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-ot 3760  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-ec 6836  df-qs 6840  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-word 11643  df-concat 11644  df-s1 11645  df-substr 11646  df-splice 11647  df-s2 11732  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-0g 13647  df-gsum 13648  df-imas 13654  df-divs 13655  df-mnd 14610  df-submnd 14659  df-frmd 14714  df-grp 14732  df-minusg 14733  df-efg 15261  df-frgp 15262
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