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Theorem frmdgsum 14500
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 4858 . . . . . . 7  |-  ( x  =  (/)  ->  ( U  o.  x )  =  ( U  o.  (/) ) )
2 co02 5202 . . . . . . 7  |-  ( U  o.  (/) )  =  (/)
31, 2syl6eq 2344 . . . . . 6  |-  ( x  =  (/)  ->  ( U  o.  x )  =  (/) )
43oveq2d 5890 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  ( U  o.  x
) )  =  ( M  gsumg  (/) ) )
5 id 19 . . . . 5  |-  ( x  =  (/)  ->  x  =  (/) )
64, 5eqeq12d 2310 . . . 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 4858 . . . . . 6  |-  ( x  =  y  ->  ( U  o.  x )  =  ( U  o.  y ) )
98oveq2d 5890 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  y
) ) )
10 id 19 . . . . 5  |-  ( x  =  y  ->  x  =  y )
119, 10eqeq12d 2310 . . . 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 4858 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  ( U  o.  x
)  =  ( U  o.  ( y concat  <" z "> )
) )
1413oveq2d 5890 . . . . 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 2310 . . . 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 4858 . . . . . 6  |-  ( x  =  W  ->  ( U  o.  x )  =  ( U  o.  W ) )
1918oveq2d 5890 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  ( U  o.  x
) )  =  ( M  gsumg  ( U  o.  W
) ) )
20 id 19 . . . . 5  |-  ( x  =  W  ->  x  =  W )
2119, 20eqeq12d 2310 . . . 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 14498 . . . . 5  |-  (/)  =  ( 0g `  M )
2524gsum0 14473 . . . 4  |-  ( M 
gsumg  (/) )  =  (/)
2625a1i 10 . . 3  |-  ( I  e.  V  ->  ( M  gsumg  (/) )  =  (/) )
27 oveq1 5881 . . . . . 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 11458 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e. Word  I )
31 frmdgsum.u . . . . . . . . . . . . 13  |-  U  =  (varFMnd `  I )
3231vrmdf 14496 . . . . . . . . . . . 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 11506 . . . . . . . . . . 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 11504 . . . . . . . . . . . . 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 14495 . . . . . . . . . . . . . 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 11456 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" ( U `  z ) ">  =  <" <" z "> "> )
4137, 40eqtrd 2328 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  <" z "> )  =  <" <" z "> "> )
4241oveq2d 5890 . . . . . . . . . 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 2328 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  ( y concat  <" z "> ) )  =  ( ( U  o.  y
) concat  <" <" z "> "> )
)
4443oveq2d 5890 . . . . . . . 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 14497 . . . . . . . . . . 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 11502 . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . 14  |-  ( Base `  M )  =  (
Base `  M )
5023, 49frmdbas 14490 . . . . . . . . . . . . 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 11440 . . . . . . . . . . . 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 2373 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  ( U  o.  y )  e. Word  ( Base `  M
) )
5530, 51eleqtrrd 2373 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" z ">  e.  ( Base `  M ) )
5655s1cld 11458 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  ( y  e. Word  I  /\  z  e.  I
) )  ->  <" <" z "> ">  e. Word  ( Base `  M
) )
57 eqid 2296 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
5849, 57gsumccat 14480 . . . . . . . . . 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 14478 . . . . . . . . . . . 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 5890 . . . . . . . . . 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 14479 . . . . . . . . . . . 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 14493 . . . . . . . . . . 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 2328 . . . . . . . . 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 2328 . . . . . . . 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 2328 . . . . . . 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 2304 . . . . . 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 11493 . 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 1632    e. wcel 1696   (/)c0 3468    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874  Word cword 11419   concat cconcat 11420   <"cs1 11421   Basecbs 13164   +g cplusg 13224    gsumg cgsu 13417   Mndcmnd 14377  freeMndcfrmd 14485  varFMndcvrmd 14486
This theorem is referenced by:  frmdss2  14501  frmdup3  14504  frgpup3lem  15102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432  df-frmd 14487  df-vrmd 14488
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