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Theorem frmdgsum 14799
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 5023 . . . . . . 7  |-  ( x  =  (/)  ->  ( U  o.  x )  =  ( U  o.  (/) ) )
2 co02 5375 . . . . . . 7  |-  ( U  o.  (/) )  =  (/)
31, 2syl6eq 2483 . . . . . 6  |-  ( x  =  (/)  ->  ( U  o.  x )  =  (/) )
43oveq2d 6089 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  ( U  o.  x
) )  =  ( M  gsumg  (/) ) )
5 id 20 . . . . 5  |-  ( x  =  (/)  ->  x  =  (/) )
64, 5eqeq12d 2449 . . . 4  |-  ( x  =  (/)  ->  ( ( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  (/) )  =  (/) ) )
76imbi2d 308 . . 3  |-  ( x  =  (/)  ->  ( ( I  e.  V  -> 
( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) ) ) )
8 coeq2 5023 . . . . . 6  |-  ( x  =  y  ->  ( U  o.  x )  =  ( U  o.  y ) )
98oveq2d 6089 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  y
) ) )
10 id 20 . . . . 5  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2449 . . . 4  |-  ( x  =  y  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  y
) )  =  y ) )
1211imbi2d 308 . . 3  |-  ( x  =  y  ->  (
( I  e.  V  ->  ( M  gsumg  ( U  o.  x
) )  =  x )  <->  ( I  e.  V  ->  ( M  gsumg  ( U  o.  y ) )  =  y ) ) )
13 coeq2 5023 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  ( U  o.  x
)  =  ( U  o.  ( y concat  <" z "> )
) )
1413oveq2d 6089 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  (
y concat  <" z "> ) ) ) )
15 id 20 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  x  =  ( y concat  <" z "> ) )
1614, 15eqeq12d 2449 . . . 4  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( M  gsumg  ( U  o.  x ) )  =  x  <->  ( M  gsumg  ( U  o.  ( y concat  <" z "> ) ) )  =  ( y concat  <" z "> ) ) )
1716imbi2d 308 . . 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 5023 . . . . . 6  |-  ( x  =  W  ->  ( U  o.  x )  =  ( U  o.  W ) )
1918oveq2d 6089 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  W
) ) )
20 id 20 . . . . 5  |-  ( x  =  W  ->  x  =  W )
2119, 20eqeq12d 2449 . . . 4  |-  ( x  =  W  ->  (
( M  gsumg  ( U  o.  x
) )  =  x  <-> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
2221imbi2d 308 . . 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 14797 . . . . 5  |-  (/)  =  ( 0g `  M )
2524gsum0 14772 . . . 4  |-  ( M 
gsumg  (/) )  =  (/)
2625a1i 11 . . 3  |-  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) )
27 oveq1 6080 . . . . . 6  |-  ( ( M  gsumg  ( U  o.  y
) )  =  y  ->  ( ( M 
gsumg  ( U  o.  y
) ) concat  <" z "> )  =  ( y concat  <" z "> ) )
28 simprl 733 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  y  e. Word  I )
29 simprr 734 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  z  e.  I )
3029s1cld 11748 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e. Word  I )
31 frmdgsum.u . . . . . . . . . . . . 13  |-  U  =  (varFMnd `  I )
3231vrmdf 14795 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  U : I -->Word  I )
3332adantr 452 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  U : I -->Word  I )
34 ccatco 11796 . . . . . . . . . . 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 1184 . . . . . . . . . 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 11794 . . . . . . . . . . . . 13  |-  ( ( z  e.  I  /\  U : I -->Word  I )  ->  ( U  o.  <" z "> )  =  <" ( U `
 z ) "> )
3729, 33, 36syl2anc 643 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" ( U `  z
) "> )
3831vrmdval 14794 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  z  e.  I )  ->  ( U `  z
)  =  <" z "> )
3938adantrl 697 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U `  z )  =  <" z "> )
4039s1eqd 11746 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" ( U `  z ) ">  =  <" <" z "> "> )
4137, 40eqtrd 2467 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" <" z "> "> )
4241oveq2d 6089 . . . . . . . . . 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 2467 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  ( y concat  <" z "> ) )  =  ( ( U  o.  y
) concat  <" <" z "> "> )
)
4443oveq2d 6089 . . . . . . . 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 14796 . . . . . . . . . . 11  |-  ( I  e.  V  ->  M  e.  Mnd )
4645adantr 452 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  M  e.  Mnd )
47 wrdco 11792 . . . . . . . . . . . 12  |-  ( ( y  e. Word  I  /\  U : I -->Word  I )  ->  ( U  o.  y
)  e. Word Word  I )
4828, 33, 47syl2anc 643 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word Word  I )
49 eqid 2435 . . . . . . . . . . . . . 14  |-  ( Base `  M )  =  (
Base `  M )
5023, 49frmdbas 14789 . . . . . . . . . . . . 13  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
5150adantr 452 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( Base `  M )  = Word 
I )
52 wrdeq 11730 . . . . . . . . . . . 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 2512 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word  ( Base `  M
) )
5530, 51eleqtrrd 2512 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e.  ( Base `  M ) )
5655s1cld 11748 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" <" z "> ">  e. Word  ( Base `  M
) )
57 eqid 2435 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
5849, 57gsumccat 14779 . . . . . . . . . 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 1184 . . . . . . . . 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 14777 . . . . . . . . . . . 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 6089 . . . . . . . . . 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 14778 . . . . . . . . . . . 12  |-  ( ( M  e.  Mnd  /\  ( U  o.  y
)  e. Word  ( Base `  M ) )  -> 
( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6446, 54, 63syl2anc 643 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( M  gsumg  ( U  o.  y
) )  e.  (
Base `  M )
)
6523, 49, 57frmdadd 14792 . . . . . . . . . . 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 643 . . . . . . . . . 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 2467 . . . . . . . . 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 2467 . . . . . . . 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 2467 . . . . . . 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 2443 . . . . . 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 213 . . . . 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 425 . . . 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 24 . . 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 11783 . 2  |-  ( W  e. Word  I  ->  (
I  e.  V  -> 
( M  gsumg  ( U  o.  W
) )  =  W ) )
7574impcom 420 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 359    = wceq 1652    e. wcel 1725   (/)c0 3620    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073  Word cword 11709   concat cconcat 11710   <"cs1 11711   Basecbs 13461   +g cplusg 13521    gsumg cgsu 13716   Mndcmnd 14676  freeMndcfrmd 14784  varFMndcvrmd 14785
This theorem is referenced by:  frmdss2  14800  frmdup3  14803  frgpup3lem  15401
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-word 11715  df-concat 11716  df-s1 11717  df-substr 11718  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mnd 14682  df-submnd 14731  df-frmd 14786  df-vrmd 14787
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