MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgpmhm Structured version   Unicode version

Theorem frgpmhm 16260
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 6926 . . . . 5  |-  2o  e.  On
2 xpexg 6505 . . . . 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 15535 . . . 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 16257 . . . 4  |-  ( I  e.  V  ->  G  e.  Grp )
9 grpmnd 15548 . . . 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 15528 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  W  = Word  ( I  X.  2o ) )
14 wrdexg 12242 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
15 fvi 5746 . . . . . . . . . 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 2476 . . . . . . . 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 2508 . . . . . 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 2441 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
23 eqid 2441 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
247, 21, 22, 23frgpeccl 16256 . . . . 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 5865 . . 3  |-  ( I  e.  V  ->  F : W --> ( Base `  G
) )
2822, 21efger 16213 . . . . . . . 8  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
29 ereq2 7107 . . . . . . . . 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 5699 . . . . . . . 8  |-  ( Base `  M )  e.  _V
3412, 33eqeltri 2511 . . . . . . 7  |-  W  e. 
_V
3534a1i 11 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  W  e.  _V )
3632, 35, 26divsfval 14483 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a concat  b ) )  =  [
( a concat  b ) ]  .~  )
37 eqid 2441 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
384, 12, 37frmdadd 15531 . . . . . . 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 5693 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( F `  (
a concat  b ) ) )
4132, 35, 26divsfval 14483 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  a )  =  [ a ]  .~  )
4232, 35, 26divsfval 14483 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  b )  =  [ b ]  .~  )
4341, 42oveq12d 6107 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  ( [ a ]  .~  ( +g  `  G
) [ b ]  .~  ) )
4418eleq2d 2508 . . . . . . . . 9  |-  ( I  e.  V  ->  (
a  e.  W  <->  a  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4518eleq2d 2508 . . . . . . . . 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 2441 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
4922, 7, 21, 48frgpadd 16258 . . . . . . 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 2473 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  [ ( a concat  b
) ]  .~  )
5236, 40, 513eqtr4d 2483 . . . 4  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G ) ( F `
 b ) ) )
5352ralrimivva 2806 . . 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 14483 . . . 4  |-  ( I  e.  V  ->  ( F `  (/) )  =  [ (/) ]  .~  )
567, 21frgp0 16255 . . . . 5  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
5756simprd 463 . . . 4  |-  ( I  e.  V  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
5855, 57eqtrd 2473 . . 3  |-  ( I  e.  V  ->  ( F `  (/) )  =  ( 0g `  G
) )
5927, 53, 583jca 1168 . 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 15536 . . 3  |-  (/)  =  ( 0g `  M )
61 eqid 2441 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
6212, 23, 37, 48, 60, 61ismhm 15464 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2713   _Vcvv 2970   (/)c0 3635    e. cmpt 4348    _I cid 4629   Oncon0 4717    X. cxp 4836   -->wf 5412   ` cfv 5416  (class class class)co 6089   2oc2o 6912    Er wer 7096   [cec 7097  Word cword 12219   concat cconcat 12221   Basecbs 14172   +g cplusg 14236   0gc0g 14376   Mndcmnd 15407   Grpcgrp 15408   MndHom cmhm 15460  freeMndcfrmd 15523   ~FG cefg 16201  freeGrpcfrgp 16202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-ot 3884  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-ec 7101  df-qs 7105  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-substr 12231  df-splice 12232  df-reverse 12233  df-s2 12473  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-0g 14378  df-imas 14444  df-divs 14445  df-mnd 15413  df-mhm 15462  df-frmd 15525  df-grp 15543  df-efg 16204  df-frgp 16205
This theorem is referenced by:  frgpup3lem  16272
  Copyright terms: Public domain W3C validator