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Theorem frmdgsum 15853
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 5159 . . . . . . 7  |-  ( x  =  (/)  ->  ( U  o.  x )  =  ( U  o.  (/) ) )
2 co02 5519 . . . . . . 7  |-  ( U  o.  (/) )  =  (/)
31, 2syl6eq 2524 . . . . . 6  |-  ( x  =  (/)  ->  ( U  o.  x )  =  (/) )
43oveq2d 6298 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  ( U  o.  x
) )  =  ( M  gsumg  (/) ) )
5 id 22 . . . . 5  |-  ( x  =  (/)  ->  x  =  (/) )
64, 5eqeq12d 2489 . . . 4  |-  ( x  =  (/)  ->  ( ( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  (/) )  =  (/) ) )
76imbi2d 316 . . 3  |-  ( x  =  (/)  ->  ( ( I  e.  V  -> 
( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) ) ) )
8 coeq2 5159 . . . . . 6  |-  ( x  =  y  ->  ( U  o.  x )  =  ( U  o.  y ) )
98oveq2d 6298 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  y
) ) )
10 id 22 . . . . 5  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2489 . . . 4  |-  ( x  =  y  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  y
) )  =  y ) )
1211imbi2d 316 . . 3  |-  ( x  =  y  ->  (
( I  e.  V  ->  ( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  ( U  o.  y ) )  =  y ) ) )
13 coeq2 5159 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  ( U  o.  x
)  =  ( U  o.  ( y concat  <" z "> )
) )
1413oveq2d 6298 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) ) )
15 id 22 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  x  =  ( y concat  <" z "> ) )
1614, 15eqeq12d 2489 . . . 4  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( M  gsumg  ( U  o.  x ) )  =  x  <->  ( M  gsumg  ( U  o.  ( y concat  <" z "> ) ) )  =  ( y concat  <" z "> ) ) )
1716imbi2d 316 . . 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 5159 . . . . . 6  |-  ( x  =  W  ->  ( U  o.  x )  =  ( U  o.  W ) )
1918oveq2d 6298 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  W
) ) )
20 id 22 . . . . 5  |-  ( x  =  W  ->  x  =  W )
2119, 20eqeq12d 2489 . . . 4  |-  ( x  =  W  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
2221imbi2d 316 . . 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 15851 . . . . 5  |-  (/)  =  ( 0g `  M )
2524gsum0 15823 . . . 4  |-  ( M 
gsumg  (/) )  =  (/)
2625a1i 11 . . 3  |-  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) )
27 oveq1 6289 . . . . . 6  |-  ( ( M  gsumg  ( U  o.  y
) )  =  y  ->  ( ( M 
gsumg  ( U  o.  y
) ) concat  <" z "> )  =  ( y concat  <" z "> ) )
28 simprl 755 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  y  e. Word  I )
29 simprr 756 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  z  e.  I )
3029s1cld 12574 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e. Word  I )
31 frmdgsum.u . . . . . . . . . . . . 13  |-  U  =  (varFMnd `  I )
3231vrmdf 15849 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  U : I -->Word  I )
3332adantr 465 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  U : I -->Word  I )
34 ccatco 12760 . . . . . . . . . . 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 1228 . . . . . . . . . 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 12758 . . . . . . . . . . . . 13  |-  ( ( z  e.  I  /\  U : I -->Word  I )  ->  ( U  o.  <" z "> )  =  <" ( U `
 z ) "> )
3729, 33, 36syl2anc 661 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" ( U `  z
) "> )
3831vrmdval 15848 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  z  e.  I )  ->  ( U `  z
)  =  <" z "> )
3938adantrl 715 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U `  z )  =  <" z "> )
4039s1eqd 12572 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" ( U `  z ) ">  =  <" <" z "> "> )
4137, 40eqtrd 2508 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" <" z "> "> )
4241oveq2d 6298 . . . . . . . . . 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 2508 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  ( y concat  <" z "> ) )  =  ( ( U  o.  y
) concat  <" <" z "> "> )
)
4443oveq2d 6298 . . . . . . . 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 15850 . . . . . . . . . . 11  |-  ( I  e.  V  ->  M  e.  Mnd )
4645adantr 465 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  M  e.  Mnd )
47 wrdco 12756 . . . . . . . . . . . 12  |-  ( ( y  e. Word  I  /\  U : I -->Word  I )  ->  ( U  o.  y
)  e. Word Word  I )
4828, 33, 47syl2anc 661 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word Word  I )
49 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  M )  =  (
Base `  M )
5023, 49frmdbas 15843 . . . . . . . . . . . . 13  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
5150adantr 465 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( Base `  M )  = Word 
I )
52 wrdeq 12526 . . . . . . . . . . . 12  |-  ( (
Base `  M )  = Word  I  -> Word  ( Base `  M
)  = Word Word  I )
5351, 52syl 16 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  -> Word  ( Base `  M )  = Word Word  I )
5448, 53eleqtrrd 2558 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word  ( Base `  M
) )
5530, 51eleqtrrd 2558 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e.  ( Base `  M ) )
5655s1cld 12574 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" <" z "> ">  e. Word  ( Base `  M
) )
57 eqid 2467 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
5849, 57gsumccat 15832 . . . . . . . . . 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 1228 . . . . . . . . 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 15830 . . . . . . . . . . . 12  |-  ( <" z ">  e.  ( Base `  M
)  ->  ( M  gsumg  <" <" z "> "> )  =  <" z "> )
6155, 60syl 16 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg 
<" <" z "> "> )  =  <" z "> )
6261oveq2d 6298 . . . . . . . . . 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 15831 . . . . . . . . . . . 12  |-  ( ( M  e.  Mnd  /\  ( U  o.  y
)  e. Word  ( Base `  M ) )  -> 
( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6446, 54, 63syl2anc 661 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6523, 49, 57frmdadd 15846 . . . . . . . . . . 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 661 . . . . . . . . . 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 2508 . . . . . . . . 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 2508 . . . . . . . 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 2508 . . . . . . 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 2469 . . . . . 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 221 . . . . 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 435 . . . 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 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 concat  <" z "> ) ) )  =  ( y concat  <" z "> ) ) ) )
747, 12, 17, 22, 26, 73wrdind 12661 . 2  |-  ( W  e. Word  I  ->  (
I  e.  V  -> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
7574impcom 430 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 369    = wceq 1379    e. wcel 1767   (/)c0 3785    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282  Word cword 12496   concat cconcat 12498   <"cs1 12499   Basecbs 14486   +g cplusg 14551    gsumg cgsu 14692   Mndcmnd 15722  freeMndcfrmd 15838  varFMndcvrmd 15839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-gsum 14694  df-mnd 15728  df-submnd 15778  df-frmd 15840  df-vrmd 15841
This theorem is referenced by:  frmdss2  15854  frmdup3  15857  frgpup3lem  16591
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