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Theorem swrdccatid 12681
Description: A prefix of a concatenation of length of the first concatenated word is the first word itself. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
Assertion
Ref Expression
swrdccatid  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  (
( A concat  B ) substr  <.
0 ,  N >. )  =  A )

Proof of Theorem swrdccatid
StepHypRef Expression
1 3simpa 993 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  ( A  e. Word  V  /\  B  e. Word  V ) )
2 lencl 12524 . . . . 5  |-  ( A  e. Word  V  ->  ( # `
 A )  e. 
NN0 )
3 lencl 12524 . . . . . 6  |-  ( B  e. Word  V  ->  ( # `
 B )  e. 
NN0 )
4 simplr 754 . . . . . . . . 9  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  ( # `
 A )  e. 
NN0 )
5 eleq1 2539 . . . . . . . . . 10  |-  ( N  =  ( # `  A
)  ->  ( N  e.  NN0  <->  ( # `  A
)  e.  NN0 )
)
65adantl 466 . . . . . . . . 9  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  ( N  e.  NN0  <->  ( # `  A
)  e.  NN0 )
)
74, 6mpbird 232 . . . . . . . 8  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  N  e.  NN0 )
8 nn0addcl 10827 . . . . . . . . . 10  |-  ( ( ( # `  A
)  e.  NN0  /\  ( # `  B )  e.  NN0 )  -> 
( ( # `  A
)  +  ( # `  B ) )  e. 
NN0 )
98ancoms 453 . . . . . . . . 9  |-  ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  -> 
( ( # `  A
)  +  ( # `  B ) )  e. 
NN0 )
109adantr 465 . . . . . . . 8  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  (
( # `  A )  +  ( # `  B
) )  e.  NN0 )
11 nn0re 10800 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  NN0  ->  ( # `  A
)  e.  RR )
1211anim1i 568 . . . . . . . . . . . 12  |-  ( ( ( # `  A
)  e.  NN0  /\  ( # `  B )  e.  NN0 )  -> 
( ( # `  A
)  e.  RR  /\  ( # `  B )  e.  NN0 ) )
1312ancoms 453 . . . . . . . . . . 11  |-  ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  -> 
( ( # `  A
)  e.  RR  /\  ( # `  B )  e.  NN0 ) )
14 nn0addge1 10838 . . . . . . . . . . 11  |-  ( ( ( # `  A
)  e.  RR  /\  ( # `  B )  e.  NN0 )  -> 
( # `  A )  <_  ( ( # `  A )  +  (
# `  B )
) )
1513, 14syl 16 . . . . . . . . . 10  |-  ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  -> 
( # `  A )  <_  ( ( # `  A )  +  (
# `  B )
) )
1615adantr 465 . . . . . . . . 9  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  ( # `
 A )  <_ 
( ( # `  A
)  +  ( # `  B ) ) )
17 breq1 4450 . . . . . . . . . 10  |-  ( N  =  ( # `  A
)  ->  ( N  <_  ( ( # `  A
)  +  ( # `  B ) )  <->  ( # `  A
)  <_  ( ( # `
 A )  +  ( # `  B
) ) ) )
1817adantl 466 . . . . . . . . 9  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  ( N  <_  ( ( # `  A )  +  (
# `  B )
)  <->  ( # `  A
)  <_  ( ( # `
 A )  +  ( # `  B
) ) ) )
1916, 18mpbird 232 . . . . . . . 8  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  N  <_  ( ( # `  A
)  +  ( # `  B ) ) )
20 elfz2nn0 11764 . . . . . . . 8  |-  ( N  e.  ( 0 ... ( ( # `  A
)  +  ( # `  B ) ) )  <-> 
( N  e.  NN0  /\  ( ( # `  A
)  +  ( # `  B ) )  e. 
NN0  /\  N  <_  ( ( # `  A
)  +  ( # `  B ) ) ) )
217, 10, 19, 20syl3anbrc 1180 . . . . . . 7  |-  ( ( ( ( # `  B
)  e.  NN0  /\  ( # `  A )  e.  NN0 )  /\  N  =  ( # `  A
) )  ->  N  e.  ( 0 ... (
( # `  A )  +  ( # `  B
) ) ) )
2221exp31 604 . . . . . 6  |-  ( (
# `  B )  e.  NN0  ->  ( ( # `
 A )  e. 
NN0  ->  ( N  =  ( # `  A
)  ->  N  e.  ( 0 ... (
( # `  A )  +  ( # `  B
) ) ) ) ) )
233, 22syl 16 . . . . 5  |-  ( B  e. Word  V  ->  (
( # `  A )  e.  NN0  ->  ( N  =  ( # `  A
)  ->  N  e.  ( 0 ... (
( # `  A )  +  ( # `  B
) ) ) ) ) )
242, 23syl5com 30 . . . 4  |-  ( A  e. Word  V  ->  ( B  e. Word  V  ->  ( N  =  ( # `  A
)  ->  N  e.  ( 0 ... (
( # `  A )  +  ( # `  B
) ) ) ) ) )
25243imp 1190 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  N  e.  ( 0 ... (
( # `  A )  +  ( # `  B
) ) ) )
26 eqid 2467 . . . 4  |-  ( # `  A )  =  (
# `  A )
2726swrdccat3a 12678 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( N  e.  ( 0 ... ( (
# `  A )  +  ( # `  B
) ) )  -> 
( ( A concat  B
) substr  <. 0 ,  N >. )  =  if ( N  <_  ( # `  A
) ,  ( A substr  <. 0 ,  N >. ) ,  ( A concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >.
) ) ) ) )
281, 25, 27sylc 60 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  (
( A concat  B ) substr  <.
0 ,  N >. )  =  if ( N  <_  ( # `  A
) ,  ( A substr  <. 0 ,  N >. ) ,  ( A concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >.
) ) ) )
292, 11syl 16 . . . . . 6  |-  ( A  e. Word  V  ->  ( # `
 A )  e.  RR )
3029leidd 10115 . . . . 5  |-  ( A  e. Word  V  ->  ( # `
 A )  <_ 
( # `  A ) )
31303ad2ant1 1017 . . . 4  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  ( # `
 A )  <_ 
( # `  A ) )
32 breq1 4450 . . . . 5  |-  ( N  =  ( # `  A
)  ->  ( N  <_  ( # `  A
)  <->  ( # `  A
)  <_  ( # `  A
) ) )
33323ad2ant3 1019 . . . 4  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  ( N  <_  ( # `  A
)  <->  ( # `  A
)  <_  ( # `  A
) ) )
3431, 33mpbird 232 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  N  <_  ( # `  A
) )
35 iftrue 3945 . . 3  |-  ( N  <_  ( # `  A
)  ->  if ( N  <_  ( # `  A
) ,  ( A substr  <. 0 ,  N >. ) ,  ( A concat  ( B substr  <. 0 ,  ( N  -  ( # `  A ) ) >.
) ) )  =  ( A substr  <. 0 ,  N >. ) )
3634, 35syl 16 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  if ( N  <_  ( # `  A ) ,  ( A substr  <. 0 ,  N >. ) ,  ( A concat 
( B substr  <. 0 ,  ( N  -  ( # `
 A ) )
>. ) ) )  =  ( A substr  <. 0 ,  N >. ) )
37 swrdid 12611 . . . 4  |-  ( A  e. Word  V  ->  ( A substr  <. 0 ,  (
# `  A ) >. )  =  A )
38373ad2ant1 1017 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  ( A substr  <. 0 ,  (
# `  A ) >. )  =  A )
39 opeq2 4214 . . . . . 6  |-  ( N  =  ( # `  A
)  ->  <. 0 ,  N >.  =  <. 0 ,  ( # `  A
) >. )
4039oveq2d 6298 . . . . 5  |-  ( N  =  ( # `  A
)  ->  ( A substr  <.
0 ,  N >. )  =  ( A substr  <. 0 ,  ( # `  A
) >. ) )
4140eqeq1d 2469 . . . 4  |-  ( N  =  ( # `  A
)  ->  ( ( A substr  <. 0 ,  N >. )  =  A  <->  ( A substr  <.
0 ,  ( # `  A ) >. )  =  A ) )
42413ad2ant3 1019 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  (
( A substr  <. 0 ,  N >. )  =  A  <-> 
( A substr  <. 0 ,  ( # `  A
) >. )  =  A ) )
4338, 42mpbird 232 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  ( A substr  <. 0 ,  N >. )  =  A )
4428, 36, 433eqtrd 2512 1  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A
) )  ->  (
( A concat  B ) substr  <.
0 ,  N >. )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ifcif 3939   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488    + caddc 9491    <_ cle 9625    - cmin 9801   NN0cn0 10791   ...cfz 11668   #chash 12369  Word cword 12496   concat cconcat 12498   substr csubstr 12500
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-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-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-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-concat 12506  df-substr 12508
This theorem is referenced by:  ccats1swrdeqbi  12682  clwlkisclwwlk2  24466  clwlkfoclwwlk  24521  numclwlk1lem2foa  24768  numclwlk1lem2fo  24772
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