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Theorem frgpmhm 15090
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 6503 . . . . 5  |-  2o  e.  On
2 xpexg 4816 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
31, 2mpan2 652 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
4 frgpmhm.m . . . . 5  |-  M  =  (freeMnd `  ( I  X.  2o ) )
54frmdmnd 14497 . . . 4  |-  ( ( I  X.  2o )  e.  _V  ->  M  e.  Mnd )
63, 5syl 15 . . 3  |-  ( I  e.  V  ->  M  e.  Mnd )
7 frgpmhm.g . . . . 5  |-  G  =  (freeGrp `  I )
87frgpgrp 15087 . . . 4  |-  ( I  e.  V  ->  G  e.  Grp )
9 grpmnd 14510 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
108, 9syl 15 . . 3  |-  ( I  e.  V  ->  G  e.  Mnd )
116, 10jca 518 . 2  |-  ( I  e.  V  ->  ( M  e.  Mnd  /\  G  e.  Mnd ) )
12 frgpmhm.w . . . . . . . . . 10  |-  W  =  ( Base `  M
)
134, 12frmdbas 14490 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  W  = Word  ( I  X.  2o ) )
14 wrdexg 11441 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
15 fvi 5595 . . . . . . . . . 10  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1614, 15syl 15 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1713, 16eqtr4d 2331 . . . . . . . 8  |-  ( ( I  X.  2o )  e.  _V  ->  W  =  (  _I  ` Word  ( I  X.  2o ) ) )
183, 17syl 15 . . . . . . 7  |-  ( I  e.  V  ->  W  =  (  _I  ` Word  ( I  X.  2o ) ) )
1918eleq2d 2363 . . . . . 6  |-  ( I  e.  V  ->  (
x  e.  W  <->  x  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
2019biimpa 470 . . . . 5  |-  ( ( I  e.  V  /\  x  e.  W )  ->  x  e.  (  _I 
` Word  ( I  X.  2o ) ) )
21 frgpmhm.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
22 eqid 2296 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
23 eqid 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
247, 21, 22, 23frgpeccl 15086 . . . . 5  |-  ( x  e.  (  _I  ` Word  ( I  X.  2o ) )  ->  [ x ]  .~  e.  ( Base `  G ) )
2520, 24syl 15 . . . 4  |-  ( ( I  e.  V  /\  x  e.  W )  ->  [ x ]  .~  e.  ( Base `  G
) )
26 frgpmhm.f . . . 4  |-  F  =  ( x  e.  W  |->  [ x ]  .~  )
2725, 26fmptd 5700 . . 3  |-  ( I  e.  V  ->  F : W --> ( Base `  G
) )
2822, 21efger 15043 . . . . . . . 8  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
29 ereq2 6684 . . . . . . . . 9  |-  ( W  =  (  _I  ` Word  ( I  X.  2o ) )  ->  (  .~  Er  W  <->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) ) )
3018, 29syl 15 . . . . . . . 8  |-  ( I  e.  V  ->  (  .~  Er  W  <->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) ) )
3128, 30mpbiri 224 . . . . . . 7  |-  ( I  e.  V  ->  .~  Er  W )
3231adantr 451 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  .~  Er  W )
33 fvex 5555 . . . . . . . 8  |-  ( Base `  M )  e.  _V
3412, 33eqeltri 2366 . . . . . . 7  |-  W  e. 
_V
3534a1i 10 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  W  e.  _V )
3632, 35, 26divsfval 13465 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a concat  b ) )  =  [
( a concat  b ) ]  .~  )
37 eqid 2296 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
384, 12, 37frmdadd 14493 . . . . . . 7  |-  ( ( a  e.  W  /\  b  e.  W )  ->  ( a ( +g  `  M ) b )  =  ( a concat  b
) )
3938adantl 452 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
a ( +g  `  M
) b )  =  ( a concat  b ) )
4039fveq2d 5545 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( F `  (
a concat  b ) ) )
4132, 35, 26divsfval 13465 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  a )  =  [ a ]  .~  )
4232, 35, 26divsfval 13465 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  b )  =  [ b ]  .~  )
4341, 42oveq12d 5892 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  ( [ a ]  .~  ( +g  `  G
) [ b ]  .~  ) )
4418eleq2d 2363 . . . . . . . . 9  |-  ( I  e.  V  ->  (
a  e.  W  <->  a  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4518eleq2d 2363 . . . . . . . . 9  |-  ( I  e.  V  ->  (
b  e.  W  <->  b  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4644, 45anbi12d 691 . . . . . . . 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 470 . . . . . . 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 2296 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
4922, 7, 21, 48frgpadd 15088 . . . . . . 7  |-  ( ( a  e.  (  _I 
` Word  ( I  X.  2o ) )  /\  b  e.  (  _I  ` Word  ( I  X.  2o ) ) )  ->  ( [
a ]  .~  ( +g  `  G ) [ b ]  .~  )  =  [ ( a concat  b
) ]  .~  )
5047, 49syl 15 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( [ a ]  .~  ( +g  `  G ) [ b ]  .~  )  =  [ (
a concat  b ) ]  .~  )
5143, 50eqtrd 2328 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  [ ( a concat  b
) ]  .~  )
5236, 40, 513eqtr4d 2338 . . . 4  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G ) ( F `
 b ) ) )
5352ralrimivva 2648 . . 3  |-  ( I  e.  V  ->  A. a  e.  W  A. b  e.  W  ( F `  ( a ( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G
) ( F `  b ) ) )
5434a1i 10 . . . . 5  |-  ( I  e.  V  ->  W  e.  _V )
5531, 54, 26divsfval 13465 . . . 4  |-  ( I  e.  V  ->  ( F `  (/) )  =  [ (/) ]  .~  )
567, 21frgp0 15085 . . . . 5  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
5756simprd 449 . . . 4  |-  ( I  e.  V  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
5855, 57eqtrd 2328 . . 3  |-  ( I  e.  V  ->  ( F `  (/) )  =  ( 0g `  G
) )
5927, 53, 583jca 1132 . 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 14498 . . 3  |-  (/)  =  ( 0g `  M )
61 eqid 2296 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
6212, 23, 37, 48, 60, 61ismhm 14433 . 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 645 1  |-  ( I  e.  V  ->  F  e.  ( M MndHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   (/)c0 3468    e. cmpt 4093    _I cid 4320   Oncon0 4408    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   2oc2o 6489    Er wer 6673   [cec 6674  Word cword 11419   concat cconcat 11420   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378   MndHom cmhm 14429  freeMndcfrmd 14485   ~FG cefg 15031  freeGrpcfrgp 15032
This theorem is referenced by:  frgpup3lem  15102
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-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  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-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-frmd 14487  df-grp 14505  df-efg 15034  df-frgp 15035
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