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Theorem frgpmhm 16589
Description: The "natural map" from words of the free monoid to their cosets in the free group is a surjective monoid homomorphism. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpmhm.m  |-  M  =  (freeMnd `  ( I  X.  2o ) )
frgpmhm.w  |-  W  =  ( Base `  M
)
frgpmhm.g  |-  G  =  (freeGrp `  I )
frgpmhm.r  |-  .~  =  ( ~FG  `  I )
frgpmhm.f  |-  F  =  ( x  e.  W  |->  [ x ]  .~  )
Assertion
Ref Expression
frgpmhm  |-  ( I  e.  V  ->  F  e.  ( M MndHom  G ) )
Distinct variable groups:    x, G    x, I    x, V    x, W    x,  .~
Allowed substitution hints:    F( x)    M( x)

Proof of Theorem frgpmhm
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2on 7138 . . . . 5  |-  2o  e.  On
2 xpexg 6586 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
31, 2mpan2 671 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
4 frgpmhm.m . . . . 5  |-  M  =  (freeMnd `  ( I  X.  2o ) )
54frmdmnd 15859 . . . 4  |-  ( ( I  X.  2o )  e.  _V  ->  M  e.  Mnd )
63, 5syl 16 . . 3  |-  ( I  e.  V  ->  M  e.  Mnd )
7 frgpmhm.g . . . . 5  |-  G  =  (freeGrp `  I )
87frgpgrp 16586 . . . 4  |-  ( I  e.  V  ->  G  e.  Grp )
9 grpmnd 15872 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
108, 9syl 16 . . 3  |-  ( I  e.  V  ->  G  e.  Mnd )
116, 10jca 532 . 2  |-  ( I  e.  V  ->  ( M  e.  Mnd  /\  G  e.  Mnd ) )
12 frgpmhm.w . . . . . . . . . 10  |-  W  =  ( Base `  M
)
134, 12frmdbas 15852 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  W  = Word  ( I  X.  2o ) )
14 wrdexg 12523 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
15 fvi 5924 . . . . . . . . . 10  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1614, 15syl 16 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1713, 16eqtr4d 2511 . . . . . . . 8  |-  ( ( I  X.  2o )  e.  _V  ->  W  =  (  _I  ` Word  ( I  X.  2o ) ) )
183, 17syl 16 . . . . . . 7  |-  ( I  e.  V  ->  W  =  (  _I  ` Word  ( I  X.  2o ) ) )
1918eleq2d 2537 . . . . . 6  |-  ( I  e.  V  ->  (
x  e.  W  <->  x  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
2019biimpa 484 . . . . 5  |-  ( ( I  e.  V  /\  x  e.  W )  ->  x  e.  (  _I 
` Word  ( I  X.  2o ) ) )
21 frgpmhm.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
22 eqid 2467 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
23 eqid 2467 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
247, 21, 22, 23frgpeccl 16585 . . . . 5  |-  ( x  e.  (  _I  ` Word  ( I  X.  2o ) )  ->  [ x ]  .~  e.  ( Base `  G ) )
2520, 24syl 16 . . . 4  |-  ( ( I  e.  V  /\  x  e.  W )  ->  [ x ]  .~  e.  ( Base `  G
) )
26 frgpmhm.f . . . 4  |-  F  =  ( x  e.  W  |->  [ x ]  .~  )
2725, 26fmptd 6045 . . 3  |-  ( I  e.  V  ->  F : W --> ( Base `  G
) )
2822, 21efger 16542 . . . . . . . 8  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
29 ereq2 7319 . . . . . . . . 9  |-  ( W  =  (  _I  ` Word  ( I  X.  2o ) )  ->  (  .~  Er  W  <->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) ) )
3018, 29syl 16 . . . . . . . 8  |-  ( I  e.  V  ->  (  .~  Er  W  <->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) ) )
3128, 30mpbiri 233 . . . . . . 7  |-  ( I  e.  V  ->  .~  Er  W )
3231adantr 465 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  .~  Er  W )
33 fvex 5876 . . . . . . . 8  |-  ( Base `  M )  e.  _V
3412, 33eqeltri 2551 . . . . . . 7  |-  W  e. 
_V
3534a1i 11 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  W  e.  _V )
3632, 35, 26divsfval 14802 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a concat  b ) )  =  [
( a concat  b ) ]  .~  )
37 eqid 2467 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
384, 12, 37frmdadd 15855 . . . . . . 7  |-  ( ( a  e.  W  /\  b  e.  W )  ->  ( a ( +g  `  M ) b )  =  ( a concat  b
) )
3938adantl 466 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
a ( +g  `  M
) b )  =  ( a concat  b ) )
4039fveq2d 5870 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( F `  (
a concat  b ) ) )
4132, 35, 26divsfval 14802 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  a )  =  [ a ]  .~  )
4232, 35, 26divsfval 14802 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  b )  =  [ b ]  .~  )
4341, 42oveq12d 6302 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  ( [ a ]  .~  ( +g  `  G
) [ b ]  .~  ) )
4418eleq2d 2537 . . . . . . . . 9  |-  ( I  e.  V  ->  (
a  e.  W  <->  a  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4518eleq2d 2537 . . . . . . . . 9  |-  ( I  e.  V  ->  (
b  e.  W  <->  b  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4644, 45anbi12d 710 . . . . . . . 8  |-  ( I  e.  V  ->  (
( a  e.  W  /\  b  e.  W
)  <->  ( a  e.  (  _I  ` Word  ( I  X.  2o ) )  /\  b  e.  (  _I  ` Word  ( I  X.  2o ) ) ) ) )
4746biimpa 484 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
a  e.  (  _I 
` Word  ( I  X.  2o ) )  /\  b  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
48 eqid 2467 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
4922, 7, 21, 48frgpadd 16587 . . . . . . 7  |-  ( ( a  e.  (  _I 
` Word  ( I  X.  2o ) )  /\  b  e.  (  _I  ` Word  ( I  X.  2o ) ) )  ->  ( [
a ]  .~  ( +g  `  G ) [ b ]  .~  )  =  [ ( a concat  b
) ]  .~  )
5047, 49syl 16 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( [ a ]  .~  ( +g  `  G ) [ b ]  .~  )  =  [ (
a concat  b ) ]  .~  )
5143, 50eqtrd 2508 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  [ ( a concat  b
) ]  .~  )
5236, 40, 513eqtr4d 2518 . . . 4  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G ) ( F `
 b ) ) )
5352ralrimivva 2885 . . 3  |-  ( I  e.  V  ->  A. a  e.  W  A. b  e.  W  ( F `  ( a ( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G
) ( F `  b ) ) )
5434a1i 11 . . . . 5  |-  ( I  e.  V  ->  W  e.  _V )
5531, 54, 26divsfval 14802 . . . 4  |-  ( I  e.  V  ->  ( F `  (/) )  =  [ (/) ]  .~  )
567, 21frgp0 16584 . . . . 5  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
5756simprd 463 . . . 4  |-  ( I  e.  V  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
5855, 57eqtrd 2508 . . 3  |-  ( I  e.  V  ->  ( F `  (/) )  =  ( 0g `  G
) )
5927, 53, 583jca 1176 . 2  |-  ( I  e.  V  ->  ( F : W --> ( Base `  G )  /\  A. a  e.  W  A. b  e.  W  ( F `  ( a
( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G ) ( F `
 b ) )  /\  ( F `  (/) )  =  ( 0g
`  G ) ) )
604frmd0 15860 . . 3  |-  (/)  =  ( 0g `  M )
61 eqid 2467 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
6212, 23, 37, 48, 60, 61ismhm 15788 . 2  |-  ( F  e.  ( M MndHom  G
)  <->  ( ( M  e.  Mnd  /\  G  e.  Mnd )  /\  ( F : W --> ( Base `  G )  /\  A. a  e.  W  A. b  e.  W  ( F `  ( a
( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G ) ( F `
 b ) )  /\  ( F `  (/) )  =  ( 0g
`  G ) ) ) )
6311, 59, 62sylanbrc 664 1  |-  ( I  e.  V  ->  F  e.  ( M MndHom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   (/)c0 3785    |-> cmpt 4505    _I cid 4790   Oncon0 4878    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6284   2oc2o 7124    Er wer 7308   [cec 7309  Word cword 12500   concat cconcat 12502   Basecbs 14490   +g cplusg 14555   0gc0g 14695   Mndcmnd 15726   Grpcgrp 15727   MndHom cmhm 15784  freeMndcfrmd 15847   ~FG cefg 16530  freeGrpcfrgp 16531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512  df-splice 12513  df-reverse 12514  df-s2 12776  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-0g 14697  df-imas 14763  df-divs 14764  df-mnd 15732  df-mhm 15786  df-frmd 15849  df-grp 15867  df-efg 16533  df-frgp 16534
This theorem is referenced by:  frgpup3lem  16601
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