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Theorem frgpmhm 15397
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 6732 . . . . 5  |-  2o  e.  On
2 xpexg 4989 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
31, 2mpan2 653 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
4 frgpmhm.m . . . . 5  |-  M  =  (freeMnd `  ( I  X.  2o ) )
54frmdmnd 14804 . . . 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 15394 . . . 4  |-  ( I  e.  V  ->  G  e.  Grp )
9 grpmnd 14817 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
108, 9syl 16 . . 3  |-  ( I  e.  V  ->  G  e.  Mnd )
116, 10jca 519 . 2  |-  ( I  e.  V  ->  ( M  e.  Mnd  /\  G  e.  Mnd ) )
12 frgpmhm.w . . . . . . . . . 10  |-  W  =  ( Base `  M
)
134, 12frmdbas 14797 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  W  = Word  ( I  X.  2o ) )
14 wrdexg 11739 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
15 fvi 5783 . . . . . . . . . 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 2471 . . . . . . . 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 2503 . . . . . 6  |-  ( I  e.  V  ->  (
x  e.  W  <->  x  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
2019biimpa 471 . . . . 5  |-  ( ( I  e.  V  /\  x  e.  W )  ->  x  e.  (  _I 
` Word  ( I  X.  2o ) ) )
21 frgpmhm.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
22 eqid 2436 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
23 eqid 2436 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
247, 21, 22, 23frgpeccl 15393 . . . . 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 5893 . . 3  |-  ( I  e.  V  ->  F : W --> ( Base `  G
) )
2822, 21efger 15350 . . . . . . . 8  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
29 ereq2 6913 . . . . . . . . 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 225 . . . . . . 7  |-  ( I  e.  V  ->  .~  Er  W )
3231adantr 452 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  .~  Er  W )
33 fvex 5742 . . . . . . . 8  |-  ( Base `  M )  e.  _V
3412, 33eqeltri 2506 . . . . . . 7  |-  W  e. 
_V
3534a1i 11 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  W  e.  _V )
3632, 35, 26divsfval 13772 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a concat  b ) )  =  [
( a concat  b ) ]  .~  )
37 eqid 2436 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
384, 12, 37frmdadd 14800 . . . . . . 7  |-  ( ( a  e.  W  /\  b  e.  W )  ->  ( a ( +g  `  M ) b )  =  ( a concat  b
) )
3938adantl 453 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
a ( +g  `  M
) b )  =  ( a concat  b ) )
4039fveq2d 5732 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( F `  (
a concat  b ) ) )
4132, 35, 26divsfval 13772 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  a )  =  [ a ]  .~  )
4232, 35, 26divsfval 13772 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  b )  =  [ b ]  .~  )
4341, 42oveq12d 6099 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  ( [ a ]  .~  ( +g  `  G
) [ b ]  .~  ) )
4418eleq2d 2503 . . . . . . . . 9  |-  ( I  e.  V  ->  (
a  e.  W  <->  a  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4518eleq2d 2503 . . . . . . . . 9  |-  ( I  e.  V  ->  (
b  e.  W  <->  b  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4644, 45anbi12d 692 . . . . . . . 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 471 . . . . . . 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 2436 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
4922, 7, 21, 48frgpadd 15395 . . . . . . 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 2468 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  [ ( a concat  b
) ]  .~  )
5236, 40, 513eqtr4d 2478 . . . 4  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G ) ( F `
 b ) ) )
5352ralrimivva 2798 . . 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 13772 . . . 4  |-  ( I  e.  V  ->  ( F `  (/) )  =  [ (/) ]  .~  )
567, 21frgp0 15392 . . . . 5  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
5756simprd 450 . . . 4  |-  ( I  e.  V  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
5855, 57eqtrd 2468 . . 3  |-  ( I  e.  V  ->  ( F `  (/) )  =  ( 0g `  G
) )
5927, 53, 583jca 1134 . 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 14805 . . 3  |-  (/)  =  ( 0g `  M )
61 eqid 2436 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
6212, 23, 37, 48, 60, 61ismhm 14740 . 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 646 1  |-  ( I  e.  V  ->  F  e.  ( M MndHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   (/)c0 3628    e. cmpt 4266    _I cid 4493   Oncon0 4581    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081   2oc2o 6718    Er wer 6902   [cec 6903  Word cword 11717   concat cconcat 11718   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Mndcmnd 14684   Grpcgrp 14685   MndHom cmhm 14736  freeMndcfrmd 14792   ~FG cefg 15338  freeGrpcfrgp 15339
This theorem is referenced by:  frgpup3lem  15409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-substr 11726  df-splice 11727  df-reverse 11728  df-s2 11812  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-0g 13727  df-imas 13734  df-divs 13735  df-mnd 14690  df-mhm 14738  df-frmd 14794  df-grp 14812  df-efg 15341  df-frgp 15342
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