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Theorem frgpmhm 17350
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 7198 . . . . 5  |-  2o  e.  On
2 xpexg 6607 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
31, 2mpan2 675 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
4 frgpmhm.m . . . . 5  |-  M  =  (freeMnd `  ( I  X.  2o ) )
54frmdmnd 16594 . . . 4  |-  ( ( I  X.  2o )  e.  _V  ->  M  e.  Mnd )
63, 5syl 17 . . 3  |-  ( I  e.  V  ->  M  e.  Mnd )
7 frgpmhm.g . . . . 5  |-  G  =  (freeGrp `  I )
87frgpgrp 17347 . . . 4  |-  ( I  e.  V  ->  G  e.  Grp )
9 grpmnd 16629 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
108, 9syl 17 . . 3  |-  ( I  e.  V  ->  G  e.  Mnd )
116, 10jca 534 . 2  |-  ( I  e.  V  ->  ( M  e.  Mnd  /\  G  e.  Mnd ) )
12 frgpmhm.w . . . . . . . . . 10  |-  W  =  ( Base `  M
)
134, 12frmdbas 16587 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  W  = Word  ( I  X.  2o ) )
14 wrdexg 12669 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
15 fvi 5938 . . . . . . . . . 10  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1614, 15syl 17 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1713, 16eqtr4d 2473 . . . . . . . 8  |-  ( ( I  X.  2o )  e.  _V  ->  W  =  (  _I  ` Word  ( I  X.  2o ) ) )
183, 17syl 17 . . . . . . 7  |-  ( I  e.  V  ->  W  =  (  _I  ` Word  ( I  X.  2o ) ) )
1918eleq2d 2499 . . . . . 6  |-  ( I  e.  V  ->  (
x  e.  W  <->  x  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
2019biimpa 486 . . . . 5  |-  ( ( I  e.  V  /\  x  e.  W )  ->  x  e.  (  _I 
` Word  ( I  X.  2o ) ) )
21 frgpmhm.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
22 eqid 2429 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
23 eqid 2429 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
247, 21, 22, 23frgpeccl 17346 . . . . 5  |-  ( x  e.  (  _I  ` Word  ( I  X.  2o ) )  ->  [ x ]  .~  e.  ( Base `  G ) )
2520, 24syl 17 . . . 4  |-  ( ( I  e.  V  /\  x  e.  W )  ->  [ x ]  .~  e.  ( Base `  G
) )
26 frgpmhm.f . . . 4  |-  F  =  ( x  e.  W  |->  [ x ]  .~  )
2725, 26fmptd 6061 . . 3  |-  ( I  e.  V  ->  F : W --> ( Base `  G
) )
2822, 21efger 17303 . . . . . . . 8  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
29 ereq2 7379 . . . . . . . . 9  |-  ( W  =  (  _I  ` Word  ( I  X.  2o ) )  ->  (  .~  Er  W  <->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) ) )
3018, 29syl 17 . . . . . . . 8  |-  ( I  e.  V  ->  (  .~  Er  W  <->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) ) )
3128, 30mpbiri 236 . . . . . . 7  |-  ( I  e.  V  ->  .~  Er  W )
3231adantr 466 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  .~  Er  W )
33 fvex 5891 . . . . . . . 8  |-  ( Base `  M )  e.  _V
3412, 33eqeltri 2513 . . . . . . 7  |-  W  e. 
_V
3534a1i 11 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  W  e.  _V )
3632, 35, 26divsfval 15404 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a ++  b ) )  =  [ ( a ++  b ) ]  .~  )
37 eqid 2429 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
384, 12, 37frmdadd 16590 . . . . . . 7  |-  ( ( a  e.  W  /\  b  e.  W )  ->  ( a ( +g  `  M ) b )  =  ( a ++  b ) )
3938adantl 467 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
a ( +g  `  M
) b )  =  ( a ++  b ) )
4039fveq2d 5885 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( F `  (
a ++  b ) ) )
4132, 35, 26divsfval 15404 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  a )  =  [ a ]  .~  )
4232, 35, 26divsfval 15404 . . . . . . 7  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  b )  =  [ b ]  .~  )
4341, 42oveq12d 6323 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  ( [ a ]  .~  ( +g  `  G
) [ b ]  .~  ) )
4418eleq2d 2499 . . . . . . . . 9  |-  ( I  e.  V  ->  (
a  e.  W  <->  a  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4518eleq2d 2499 . . . . . . . . 9  |-  ( I  e.  V  ->  (
b  e.  W  <->  b  e.  (  _I  ` Word  ( I  X.  2o ) ) ) )
4644, 45anbi12d 715 . . . . . . . 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 486 . . . . . . 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 2429 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
4922, 7, 21, 48frgpadd 17348 . . . . . . 7  |-  ( ( a  e.  (  _I 
` Word  ( I  X.  2o ) )  /\  b  e.  (  _I  ` Word  ( I  X.  2o ) ) )  ->  ( [
a ]  .~  ( +g  `  G ) [ b ]  .~  )  =  [ ( a ++  b ) ]  .~  )
5047, 49syl 17 . . . . . 6  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( [ a ]  .~  ( +g  `  G ) [ b ]  .~  )  =  [ (
a ++  b ) ]  .~  )
5143, 50eqtrd 2470 . . . . 5  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  (
( F `  a
) ( +g  `  G
) ( F `  b ) )  =  [ ( a ++  b ) ]  .~  )
5236, 40, 513eqtr4d 2480 . . . 4  |-  ( ( I  e.  V  /\  ( a  e.  W  /\  b  e.  W
) )  ->  ( F `  ( a
( +g  `  M ) b ) )  =  ( ( F `  a ) ( +g  `  G ) ( F `
 b ) ) )
5352ralrimivva 2853 . . 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 15404 . . . 4  |-  ( I  e.  V  ->  ( F `  (/) )  =  [ (/) ]  .~  )
567, 21frgp0 17345 . . . . 5  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
5756simprd 464 . . . 4  |-  ( I  e.  V  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
5855, 57eqtrd 2470 . . 3  |-  ( I  e.  V  ->  ( F `  (/) )  =  ( 0g `  G
) )
5927, 53, 583jca 1185 . 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 16595 . . 3  |-  (/)  =  ( 0g `  M )
61 eqid 2429 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
6212, 23, 37, 48, 60, 61ismhm 16535 . 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 668 1  |-  ( I  e.  V  ->  F  e.  ( M MndHom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087   (/)c0 3767    |-> cmpt 4484    _I cid 4764    X. cxp 4852   Oncon0 5442   -->wf 5597   ` cfv 5601  (class class class)co 6305   2oc2o 7184    Er wer 7368   [cec 7369  Word cword 12643   ++ cconcat 12645   Basecbs 15084   +g cplusg 15152   0gc0g 15297   Mndcmnd 16486   MndHom cmhm 16531  freeMndcfrmd 16582   Grpcgrp 16620   ~FG cefg 17291  freeGrpcfrgp 17292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-lsw 12652  df-concat 12653  df-s1 12654  df-substr 12655  df-splice 12656  df-reverse 12657  df-s2 12929  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-0g 15299  df-imas 15365  df-qus 15366  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-frmd 16584  df-grp 16624  df-efg 17294  df-frgp 17295
This theorem is referenced by:  frgpup3lem  17362
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