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

Theorem frmdgsum 16354
Description: Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
frmdmnd.m  |-  M  =  (freeMnd `  I )
frmdgsum.u  |-  U  =  (varFMnd `  I )
Assertion
Ref Expression
frmdgsum  |-  ( ( I  e.  V  /\  W  e. Word  I )  ->  ( M  gsumg  ( U  o.  W
) )  =  W )

Proof of Theorem frmdgsum
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coeq2 4982 . . . . . . 7  |-  ( x  =  (/)  ->  ( U  o.  x )  =  ( U  o.  (/) ) )
2 co02 5337 . . . . . . 7  |-  ( U  o.  (/) )  =  (/)
31, 2syl6eq 2459 . . . . . 6  |-  ( x  =  (/)  ->  ( U  o.  x )  =  (/) )
43oveq2d 6294 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  ( U  o.  x
) )  =  ( M  gsumg  (/) ) )
5 id 22 . . . . 5  |-  ( x  =  (/)  ->  x  =  (/) )
64, 5eqeq12d 2424 . . . 4  |-  ( x  =  (/)  ->  ( ( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  (/) )  =  (/) ) )
76imbi2d 314 . . 3  |-  ( x  =  (/)  ->  ( ( I  e.  V  -> 
( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) ) ) )
8 coeq2 4982 . . . . . 6  |-  ( x  =  y  ->  ( U  o.  x )  =  ( U  o.  y ) )
98oveq2d 6294 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  y
) ) )
10 id 22 . . . . 5  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2424 . . . 4  |-  ( x  =  y  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  y
) )  =  y ) )
1211imbi2d 314 . . 3  |-  ( x  =  y  ->  (
( I  e.  V  ->  ( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  ( U  o.  y ) )  =  y ) ) )
13 coeq2 4982 . . . . . 6  |-  ( x  =  ( y ++  <" z "> )  ->  ( U  o.  x
)  =  ( U  o.  ( y ++  <" z "> )
) )
1413oveq2d 6294 . . . . 5  |-  ( x  =  ( y ++  <" z "> )  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  (
y ++  <" z "> ) ) ) )
15 id 22 . . . . 5  |-  ( x  =  ( y ++  <" z "> )  ->  x  =  ( y ++ 
<" z "> ) )
1614, 15eqeq12d 2424 . . . 4  |-  ( x  =  ( y ++  <" z "> )  ->  ( ( M  gsumg  ( U  o.  x ) )  =  x  <->  ( M  gsumg  ( U  o.  ( y ++ 
<" z "> ) ) )  =  ( y ++  <" z "> ) ) )
1716imbi2d 314 . . 3  |-  ( x  =  ( y ++  <" z "> )  ->  ( ( I  e.  V  ->  ( M  gsumg  ( U  o.  x ) )  =  x )  <-> 
( I  e.  V  ->  ( M  gsumg  ( U  o.  (
y ++  <" z "> ) ) )  =  ( y ++  <" z "> )
) ) )
18 coeq2 4982 . . . . . 6  |-  ( x  =  W  ->  ( U  o.  x )  =  ( U  o.  W ) )
1918oveq2d 6294 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  W
) ) )
20 id 22 . . . . 5  |-  ( x  =  W  ->  x  =  W )
2119, 20eqeq12d 2424 . . . 4  |-  ( x  =  W  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
2221imbi2d 314 . . 3  |-  ( x  =  W  ->  (
( I  e.  V  ->  ( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  ( U  o.  W ) )  =  W ) ) )
23 frmdmnd.m . . . . . 6  |-  M  =  (freeMnd `  I )
2423frmd0 16352 . . . . 5  |-  (/)  =  ( 0g `  M )
2524gsum0 16229 . . . 4  |-  ( M 
gsumg  (/) )  =  (/)
2625a1i 11 . . 3  |-  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) )
27 oveq1 6285 . . . . . 6  |-  ( ( M  gsumg  ( U  o.  y
) )  =  y  ->  ( ( M 
gsumg  ( U  o.  y
) ) ++  <" z "> )  =  ( y ++  <" z "> ) )
28 simprl 756 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  y  e. Word  I )
29 simprr 758 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  z  e.  I )
3029s1cld 12669 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e. Word  I )
31 frmdgsum.u . . . . . . . . . . . . 13  |-  U  =  (varFMnd `  I )
3231vrmdf 16350 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  U : I -->Word  I )
3332adantr 463 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  U : I -->Word  I )
34 ccatco 12857 . . . . . . . . . . 11  |-  ( ( y  e. Word  I  /\  <" z ">  e. Word  I  /\  U :
I -->Word  I )  ->  ( U  o.  ( y ++  <" z "> ) )  =  ( ( U  o.  y
) ++  ( U  o.  <" z "> ) ) )
3528, 30, 33, 34syl3anc 1230 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  ( y ++  <" z "> ) )  =  ( ( U  o.  y
) ++  ( U  o.  <" z "> ) ) )
36 s1co 12855 . . . . . . . . . . . . 13  |-  ( ( z  e.  I  /\  U : I -->Word  I )  ->  ( U  o.  <" z "> )  =  <" ( U `
 z ) "> )
3729, 33, 36syl2anc 659 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" ( U `  z
) "> )
3831vrmdval 16349 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  z  e.  I )  ->  ( U `  z
)  =  <" z "> )
3938adantrl 714 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U `  z )  =  <" z "> )
4039s1eqd 12667 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" ( U `  z ) ">  =  <" <" z "> "> )
4137, 40eqtrd 2443 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" <" z "> "> )
4241oveq2d 6294 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( U  o.  y
) ++  ( U  o.  <" z "> ) )  =  ( ( U  o.  y
) ++  <" <" z "> "> )
)
4335, 42eqtrd 2443 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  ( y ++  <" z "> ) )  =  ( ( U  o.  y
) ++  <" <" z "> "> )
)
4443oveq2d 6294 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  (
y ++  <" z "> ) ) )  =  ( M  gsumg  ( ( U  o.  y ) ++ 
<" <" z "> "> )
) )
4523frmdmnd 16351 . . . . . . . . . . 11  |-  ( I  e.  V  ->  M  e.  Mnd )
4645adantr 463 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  M  e.  Mnd )
47 wrdco 12853 . . . . . . . . . . . 12  |-  ( ( y  e. Word  I  /\  U : I -->Word  I )  ->  ( U  o.  y
)  e. Word Word  I )
4828, 33, 47syl2anc 659 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word Word  I )
49 eqid 2402 . . . . . . . . . . . . . 14  |-  ( Base `  M )  =  (
Base `  M )
5023, 49frmdbas 16344 . . . . . . . . . . . . 13  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
5150adantr 463 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( Base `  M )  = Word 
I )
52 wrdeq 12616 . . . . . . . . . . . 12  |-  ( (
Base `  M )  = Word  I  -> Word  ( Base `  M
)  = Word Word  I )
5351, 52syl 17 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  -> Word  ( Base `  M )  = Word Word  I )
5448, 53eleqtrrd 2493 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word  ( Base `  M
) )
5530, 51eleqtrrd 2493 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e.  ( Base `  M ) )
5655s1cld 12669 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" <" z "> ">  e. Word  ( Base `  M
) )
57 eqid 2402 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
5849, 57gsumccat 16333 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( U  o.  y
)  e. Word  ( Base `  M )  /\  <" <" z "> ">  e. Word  (
Base `  M )
)  ->  ( M  gsumg  ( ( U  o.  y
) ++  <" <" z "> "> )
)  =  ( ( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) ( M 
gsumg  <" <" z "> "> )
) )
5946, 54, 56, 58syl3anc 1230 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( ( U  o.  y ) ++  <" <" z "> "> ) )  =  ( ( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) ( M 
gsumg  <" <" z "> "> )
) )
6049gsumws1 16331 . . . . . . . . . . . 12  |-  ( <" z ">  e.  ( Base `  M
)  ->  ( M  gsumg  <" <" z "> "> )  =  <" z "> )
6155, 60syl 17 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg 
<" <" z "> "> )  =  <" z "> )
6261oveq2d 6294 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) ( M 
gsumg  <" <" z "> "> )
)  =  ( ( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) <" z "> ) )
6349gsumwcl 16332 . . . . . . . . . . . 12  |-  ( ( M  e.  Mnd  /\  ( U  o.  y
)  e. Word  ( Base `  M ) )  -> 
( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6446, 54, 63syl2anc 659 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6523, 49, 57frmdadd 16347 . . . . . . . . . . 11  |-  ( ( ( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )  /\  <" z ">  e.  ( Base `  M ) )  -> 
( ( M  gsumg  ( U  o.  y ) ) ( +g  `  M
) <" z "> )  =  ( ( M  gsumg  ( U  o.  y
) ) ++  <" z "> ) )
6664, 55, 65syl2anc 659 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) <" z "> )  =  ( ( M  gsumg  ( U  o.  y
) ) ++  <" z "> ) )
6762, 66eqtrd 2443 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) ( M 
gsumg  <" <" z "> "> )
)  =  ( ( M  gsumg  ( U  o.  y
) ) ++  <" z "> ) )
6859, 67eqtrd 2443 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( ( U  o.  y ) ++  <" <" z "> "> ) )  =  ( ( M  gsumg  ( U  o.  y
) ) ++  <" z "> ) )
6944, 68eqtrd 2443 . . . . . . 7  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  (
y ++  <" z "> ) ) )  =  ( ( M 
gsumg  ( U  o.  y
) ) ++  <" z "> ) )
7069eqeq1d 2404 . . . . . 6  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  (
y ++  <" z "> ) ) )  =  ( y ++  <" z "> )  <->  ( ( M  gsumg  ( U  o.  y
) ) ++  <" z "> )  =  ( y ++  <" z "> ) ) )
7127, 70syl5ibr 221 . . . . 5  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  y
) )  =  y  ->  ( M  gsumg  ( U  o.  ( y ++  <" z "> )
) )  =  ( y ++  <" z "> ) ) )
7271expcom 433 . . . 4  |-  ( ( y  e. Word  I  /\  z  e.  I )  ->  ( I  e.  V  ->  ( ( M  gsumg  ( U  o.  y ) )  =  y  ->  ( M  gsumg  ( U  o.  (
y ++  <" z "> ) ) )  =  ( y ++  <" z "> )
) ) )
7372a2d 26 . . 3  |-  ( ( y  e. Word  I  /\  z  e.  I )  ->  ( ( I  e.  V  ->  ( M  gsumg  ( U  o.  y ) )  =  y )  ->  ( I  e.  V  ->  ( M  gsumg  ( U  o.  ( y ++ 
<" z "> ) ) )  =  ( y ++  <" z "> ) ) ) )
747, 12, 17, 22, 26, 73wrdind 12758 . 2  |-  ( W  e. Word  I  ->  (
I  e.  V  -> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
7574impcom 428 1  |-  ( ( I  e.  V  /\  W  e. Word  I )  ->  ( M  gsumg  ( U  o.  W
) )  =  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   (/)c0 3738    o. ccom 4827   -->wf 5565   ` cfv 5569  (class class class)co 6278  Word cword 12583   ++ cconcat 12585   <"cs1 12586   Basecbs 14841   +g cplusg 14909    gsumg cgsu 15055   Mndcmnd 16243  freeMndcfrmd 16339  varFMndcvrmd 16340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-word 12591  df-lsw 12592  df-concat 12593  df-s1 12594  df-substr 12595  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-gsum 15057  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-frmd 16341  df-vrmd 16342
This theorem is referenced by:  frmdss2  16355  frmdup3lem  16358  frgpup3lem  17119
  Copyright terms: Public domain W3C validator