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Theorem frmdgsum 14484
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 4842 . . . . . . 7  |-  ( x  =  (/)  ->  ( U  o.  x )  =  ( U  o.  (/) ) )
2 co02 5186 . . . . . . 7  |-  ( U  o.  (/) )  =  (/)
31, 2syl6eq 2331 . . . . . 6  |-  ( x  =  (/)  ->  ( U  o.  x )  =  (/) )
43oveq2d 5874 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  ( U  o.  x
) )  =  ( M  gsumg  (/) ) )
5 id 19 . . . . 5  |-  ( x  =  (/)  ->  x  =  (/) )
64, 5eqeq12d 2297 . . . 4  |-  ( x  =  (/)  ->  ( ( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  (/) )  =  (/) ) )
76imbi2d 307 . . 3  |-  ( x  =  (/)  ->  ( ( I  e.  V  -> 
( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) ) ) )
8 coeq2 4842 . . . . . 6  |-  ( x  =  y  ->  ( U  o.  x )  =  ( U  o.  y ) )
98oveq2d 5874 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  y
) ) )
10 id 19 . . . . 5  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2297 . . . 4  |-  ( x  =  y  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  y
) )  =  y ) )
1211imbi2d 307 . . 3  |-  ( x  =  y  ->  (
( I  e.  V  ->  ( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  ( U  o.  y ) )  =  y ) ) )
13 coeq2 4842 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  ( U  o.  x
)  =  ( U  o.  ( y concat  <" z "> )
) )
1413oveq2d 5874 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) ) )
15 id 19 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  x  =  ( y concat  <" z "> ) )
1614, 15eqeq12d 2297 . . . 4  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( M  gsumg  ( U  o.  x ) )  =  x  <->  ( M  gsumg  ( U  o.  ( y concat  <" z "> ) ) )  =  ( y concat  <" z "> ) ) )
1716imbi2d 307 . . 3  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( I  e.  V  ->  ( M  gsumg  ( U  o.  x ) )  =  x )  <-> 
( I  e.  V  ->  ( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) )  =  ( y concat  <" z "> )
) ) )
18 coeq2 4842 . . . . . 6  |-  ( x  =  W  ->  ( U  o.  x )  =  ( U  o.  W ) )
1918oveq2d 5874 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  W
) ) )
20 id 19 . . . . 5  |-  ( x  =  W  ->  x  =  W )
2119, 20eqeq12d 2297 . . . 4  |-  ( x  =  W  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
2221imbi2d 307 . . 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 14482 . . . . 5  |-  (/)  =  ( 0g `  M )
2524gsum0 14457 . . . 4  |-  ( M 
gsumg  (/) )  =  (/)
2625a1i 10 . . 3  |-  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) )
27 oveq1 5865 . . . . . 6  |-  ( ( M  gsumg  ( U  o.  y
) )  =  y  ->  ( ( M 
gsumg  ( U  o.  y
) ) concat  <" z "> )  =  ( y concat  <" z "> ) )
28 simprl 732 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  y  e. Word  I )
29 simprr 733 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  z  e.  I )
3029s1cld 11442 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e. Word  I )
31 frmdgsum.u . . . . . . . . . . . . 13  |-  U  =  (varFMnd `  I )
3231vrmdf 14480 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  U : I -->Word  I )
3332adantr 451 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  U : I -->Word  I )
34 ccatco 11490 . . . . . . . . . . 11  |-  ( ( y  e. Word  I  /\  <" z ">  e. Word  I  /\  U :
I -->Word  I )  ->  ( U  o.  ( y concat  <" z "> ) )  =  ( ( U  o.  y
) concat  ( U  o.  <" z "> )
) )
3528, 30, 33, 34syl3anc 1182 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  ( y concat  <" z "> ) )  =  ( ( U  o.  y
) concat  ( U  o.  <" z "> )
) )
36 s1co 11488 . . . . . . . . . . . . 13  |-  ( ( z  e.  I  /\  U : I -->Word  I )  ->  ( U  o.  <" z "> )  =  <" ( U `
 z ) "> )
3729, 33, 36syl2anc 642 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" ( U `  z
) "> )
3831vrmdval 14479 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  z  e.  I )  ->  ( U `  z
)  =  <" z "> )
3938adantrl 696 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U `  z )  =  <" z "> )
4039s1eqd 11440 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" ( U `  z ) ">  =  <" <" z "> "> )
4137, 40eqtrd 2315 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" <" z "> "> )
4241oveq2d 5874 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( U  o.  y
) concat  ( U  o.  <" z "> )
)  =  ( ( U  o.  y ) concat  <" <" z "> "> )
)
4335, 42eqtrd 2315 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  ( y concat  <" z "> ) )  =  ( ( U  o.  y
) concat  <" <" z "> "> )
)
4443oveq2d 5874 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) )  =  ( M  gsumg  ( ( U  o.  y ) concat  <" <" z "> "> )
) )
4523frmdmnd 14481 . . . . . . . . . . 11  |-  ( I  e.  V  ->  M  e.  Mnd )
4645adantr 451 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  M  e.  Mnd )
47 wrdco 11486 . . . . . . . . . . . 12  |-  ( ( y  e. Word  I  /\  U : I -->Word  I )  ->  ( U  o.  y
)  e. Word Word  I )
4828, 33, 47syl2anc 642 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word Word  I )
49 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  M )  =  (
Base `  M )
5023, 49frmdbas 14474 . . . . . . . . . . . . 13  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
5150adantr 451 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( Base `  M )  = Word 
I )
52 wrdeq 11424 . . . . . . . . . . . 12  |-  ( (
Base `  M )  = Word  I  -> Word  ( Base `  M
)  = Word Word  I )
5351, 52syl 15 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  -> Word  ( Base `  M )  = Word Word  I )
5448, 53eleqtrrd 2360 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word  ( Base `  M
) )
5530, 51eleqtrrd 2360 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e.  ( Base `  M ) )
5655s1cld 11442 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" <" z "> ">  e. Word  ( Base `  M
) )
57 eqid 2283 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
5849, 57gsumccat 14464 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( U  o.  y
)  e. Word  ( Base `  M )  /\  <" <" z "> ">  e. Word  (
Base `  M )
)  ->  ( M  gsumg  ( ( U  o.  y
) concat  <" <" z "> "> )
)  =  ( ( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) ( M 
gsumg  <" <" z "> "> )
) )
5946, 54, 56, 58syl3anc 1182 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( ( U  o.  y ) concat  <" <" z "> "> ) )  =  ( ( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) ( M 
gsumg  <" <" z "> "> )
) )
6049gsumws1 14462 . . . . . . . . . . . 12  |-  ( <" z ">  e.  ( Base `  M
)  ->  ( M  gsumg  <" <" z "> "> )  =  <" z "> )
6155, 60syl 15 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg 
<" <" z "> "> )  =  <" z "> )
6261oveq2d 5874 . . . . . . . . . 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 14463 . . . . . . . . . . . 12  |-  ( ( M  e.  Mnd  /\  ( U  o.  y
)  e. Word  ( Base `  M ) )  -> 
( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6446, 54, 63syl2anc 642 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6523, 49, 57frmdadd 14477 . . . . . . . . . . 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
) ) concat  <" z "> ) )
6664, 55, 65syl2anc 642 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  y
) ) ( +g  `  M ) <" z "> )  =  ( ( M  gsumg  ( U  o.  y
) ) concat  <" z "> ) )
6762, 66eqtrd 2315 . . . . . . . . 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
) ) concat  <" z "> ) )
6859, 67eqtrd 2315 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( ( U  o.  y ) concat  <" <" z "> "> ) )  =  ( ( M  gsumg  ( U  o.  y
) ) concat  <" z "> ) )
6944, 68eqtrd 2315 . . . . . . 7  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) )  =  ( ( M 
gsumg  ( U  o.  y
) ) concat  <" z "> ) )
7069eqeq1d 2291 . . . . . 6  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) )  =  ( y concat  <" z "> )  <->  ( ( M  gsumg  ( U  o.  y
) ) concat  <" z "> )  =  ( y concat  <" z "> ) ) )
7127, 70syl5ibr 212 . . . . 5  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  (
( M  gsumg  ( U  o.  y
) )  =  y  ->  ( M  gsumg  ( U  o.  ( y concat  <" z "> )
) )  =  ( y concat  <" z "> ) ) )
7271expcom 424 . . . 4  |-  ( ( y  e. Word  I  /\  z  e.  I )  ->  ( I  e.  V  ->  ( ( M  gsumg  ( U  o.  y ) )  =  y  ->  ( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) )  =  ( y concat  <" z "> )
) ) )
7372a2d 23 . . 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 concat  <" z "> ) ) )  =  ( y concat  <" z "> ) ) ) )
747, 12, 17, 22, 26, 73wrdind 11477 . 2  |-  ( W  e. Word  I  ->  (
I  e.  V  -> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
7574impcom 419 1  |-  ( ( I  e.  V  /\  W  e. Word  I )  ->  ( M  gsumg  ( U  o.  W
) )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   (/)c0 3455    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858  Word cword 11403   concat cconcat 11404   <"cs1 11405   Basecbs 13148   +g cplusg 13208    gsumg cgsu 13401   Mndcmnd 14361  freeMndcfrmd 14469  varFMndcvrmd 14470
This theorem is referenced by:  frmdss2  14485  frmdup3  14488  frgpup3lem  15086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-frmd 14471  df-vrmd 14472
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