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Theorem swrdccat3b 12671
Description: A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l  |-  L  =  ( # `  A
)
Assertion
Ref Expression
swrdccat3b  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( M  e.  ( 0 ... ( L  +  ( # `  B
) ) )  -> 
( ( A concat  B
) substr  <. M ,  ( L  +  ( # `  B ) ) >.
)  =  if ( L  <_  M , 
( B substr  <. ( M  -  L ) ,  ( # `  B
) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B )
) ) )

Proof of Theorem swrdccat3b
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  ( A  e. Word  V  /\  B  e. Word  V
) )
2 simpr 461 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )
3 elfz2nn0 11757 . . . . . 6  |-  ( M  e.  ( 0 ... ( L  +  (
# `  B )
) )  <->  ( M  e.  NN0  /\  ( L  +  ( # `  B
) )  e.  NN0  /\  M  <_  ( L  +  ( # `  B
) ) ) )
4 elnn0uz 11108 . . . . . . . 8  |-  ( ( L  +  ( # `  B ) )  e. 
NN0 
<->  ( L  +  (
# `  B )
)  e.  ( ZZ>= ` 
0 ) )
5 eluzfz2 11683 . . . . . . . 8  |-  ( ( L  +  ( # `  B ) )  e.  ( ZZ>= `  0 )  ->  ( L  +  (
# `  B )
)  e.  ( 0 ... ( L  +  ( # `  B ) ) ) )
64, 5sylbi 195 . . . . . . 7  |-  ( ( L  +  ( # `  B ) )  e. 
NN0  ->  ( L  +  ( # `  B ) )  e.  ( 0 ... ( L  +  ( # `  B ) ) ) )
763ad2ant2 1013 . . . . . 6  |-  ( ( M  e.  NN0  /\  ( L  +  ( # `
 B ) )  e.  NN0  /\  M  <_ 
( L  +  (
# `  B )
) )  ->  ( L  +  ( # `  B
) )  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )
83, 7sylbi 195 . . . . 5  |-  ( M  e.  ( 0 ... ( L  +  (
# `  B )
) )  ->  ( L  +  ( # `  B
) )  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )
98adantl 466 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  ( L  +  ( # `  B ) )  e.  ( 0 ... ( L  +  ( # `  B ) ) ) )
10 swrdccatin12.l . . . . . 6  |-  L  =  ( # `  A
)
1110swrdccat3 12667 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... ( L  +  ( # `  B
) ) )  /\  ( L  +  ( # `
 B ) )  e.  ( 0 ... ( L  +  (
# `  B )
) ) )  -> 
( ( A concat  B
) substr  <. M ,  ( L  +  ( # `  B ) ) >.
)  =  if ( ( L  +  (
# `  B )
)  <_  L , 
( A substr  <. M , 
( L  +  (
# `  B )
) >. ) ,  if ( L  <_  M , 
( B substr  <. ( M  -  L ) ,  ( ( L  +  ( # `  B ) )  -  L )
>. ) ,  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( ( L  +  ( # `  B
) )  -  L
) >. ) ) ) ) ) )
1211imp 429 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... ( L  +  ( # `  B
) ) )  /\  ( L  +  ( # `
 B ) )  e.  ( 0 ... ( L  +  (
# `  B )
) ) ) )  ->  ( ( A concat  B ) substr  <. M , 
( L  +  (
# `  B )
) >. )  =  if ( ( L  +  ( # `  B ) )  <_  L , 
( A substr  <. M , 
( L  +  (
# `  B )
) >. ) ,  if ( L  <_  M , 
( B substr  <. ( M  -  L ) ,  ( ( L  +  ( # `  B ) )  -  L )
>. ) ,  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( ( L  +  ( # `  B
) )  -  L
) >. ) ) ) ) )
131, 2, 9, 12syl12anc 1221 . . 3  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  ( ( A concat  B ) substr  <. M , 
( L  +  (
# `  B )
) >. )  =  if ( ( L  +  ( # `  B ) )  <_  L , 
( A substr  <. M , 
( L  +  (
# `  B )
) >. ) ,  if ( L  <_  M , 
( B substr  <. ( M  -  L ) ,  ( ( L  +  ( # `  B ) )  -  L )
>. ) ,  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( ( L  +  ( # `  B
) )  -  L
) >. ) ) ) ) )
1410swrdccat3blem 12670 . . . 4  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  ( L  +  ( # `  B ) )  <_  L )  ->  if ( L  <_  M ,  ( B substr  <.
( M  -  L
) ,  ( # `  B ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B ) )  =  ( A substr  <. M , 
( L  +  (
# `  B )
) >. ) )
15 iftrue 3938 . . . . . 6  |-  ( L  <_  M  ->  if ( L  <_  M , 
( B substr  <. ( M  -  L ) ,  ( # `  B
) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B )
)  =  ( B substr  <. ( M  -  L
) ,  ( # `  B ) >. )
)
16153ad2ant3 1014 . . . . 5  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  L  <_  M )  ->  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  (
# `  B ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B ) )  =  ( B substr  <. ( M  -  L ) ,  ( # `  B
) >. ) )
17 lencl 12515 . . . . . . . . . . . 12  |-  ( A  e. Word  V  ->  ( # `
 A )  e. 
NN0 )
1817nn0cnd 10843 . . . . . . . . . . 11  |-  ( A  e. Word  V  ->  ( # `
 A )  e.  CC )
19 lencl 12515 . . . . . . . . . . . 12  |-  ( B  e. Word  V  ->  ( # `
 B )  e. 
NN0 )
2019nn0cnd 10843 . . . . . . . . . . 11  |-  ( B  e. Word  V  ->  ( # `
 B )  e.  CC )
2110eqcomi 2473 . . . . . . . . . . . . 13  |-  ( # `  A )  =  L
2221eleq1i 2537 . . . . . . . . . . . 12  |-  ( (
# `  A )  e.  CC  <->  L  e.  CC )
23 pncan2 9816 . . . . . . . . . . . 12  |-  ( ( L  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( ( L  +  ( # `  B ) )  -  L )  =  ( # `  B
) )
2422, 23sylanb 472 . . . . . . . . . . 11  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( ( L  +  ( # `  B ) )  -  L )  =  ( # `  B
) )
2518, 20, 24syl2an 477 . . . . . . . . . 10  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( L  +  ( # `  B ) )  -  L )  =  ( # `  B
) )
2625eqcomd 2468 . . . . . . . . 9  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  B
)  =  ( ( L  +  ( # `  B ) )  -  L ) )
2726adantr 465 . . . . . . . 8  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  ( # `  B
)  =  ( ( L  +  ( # `  B ) )  -  L ) )
28273ad2ant1 1012 . . . . . . 7  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  L  <_  M )  ->  ( # `  B
)  =  ( ( L  +  ( # `  B ) )  -  L ) )
2928opeq2d 4213 . . . . . 6  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  L  <_  M )  ->  <. ( M  -  L ) ,  (
# `  B ) >.  =  <. ( M  -  L ) ,  ( ( L  +  (
# `  B )
)  -  L )
>. )
3029oveq2d 6291 . . . . 5  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  L  <_  M )  ->  ( B substr  <. ( M  -  L ) ,  ( # `  B
) >. )  =  ( B substr  <. ( M  -  L ) ,  ( ( L  +  (
# `  B )
)  -  L )
>. ) )
3116, 30eqtrd 2501 . . . 4  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  L  <_  M )  ->  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  (
# `  B ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B ) )  =  ( B substr  <. ( M  -  L ) ,  ( ( L  +  ( # `  B
) )  -  L
) >. ) )
32 iffalse 3941 . . . . . 6  |-  ( -.  L  <_  M  ->  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  ( # `  B
) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B )
)  =  ( ( A substr  <. M ,  L >. ) concat  B ) )
33323ad2ant3 1014 . . . . 5  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  (
# `  B ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B ) )  =  ( ( A substr  <. M ,  L >. ) concat  B ) )
3425adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  ( ( L  +  ( # `  B
) )  -  L
)  =  ( # `  B ) )
35343ad2ant1 1012 . . . . . . . . 9  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  ( ( L  +  ( # `  B
) )  -  L
)  =  ( # `  B ) )
3635opeq2d 4213 . . . . . . . 8  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  <. 0 ,  ( ( L  +  ( # `  B ) )  -  L )
>.  =  <. 0 ,  ( # `  B
) >. )
3736oveq2d 6291 . . . . . . 7  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  ( B substr  <.
0 ,  ( ( L  +  ( # `  B ) )  -  L ) >. )  =  ( B substr  <. 0 ,  ( # `  B
) >. ) )
38 swrdid 12602 . . . . . . . . . 10  |-  ( B  e. Word  V  ->  ( B substr  <. 0 ,  (
# `  B ) >. )  =  B )
3938adantl 466 . . . . . . . . 9  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( B substr  <. 0 ,  ( # `  B
) >. )  =  B )
4039adantr 465 . . . . . . . 8  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  ( B substr  <. 0 ,  ( # `  B
) >. )  =  B )
41403ad2ant1 1012 . . . . . . 7  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  ( B substr  <.
0 ,  ( # `  B ) >. )  =  B )
4237, 41eqtr2d 2502 . . . . . 6  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  B  =  ( B substr  <. 0 ,  ( ( L  +  ( # `  B ) )  -  L )
>. ) )
4342oveq2d 6291 . . . . 5  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  ( ( A substr  <. M ,  L >. ) concat  B )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( ( L  +  ( # `  B ) )  -  L ) >. )
) )
4433, 43eqtrd 2501 . . . 4  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  /\  -.  ( L  +  ( # `  B
) )  <_  L  /\  -.  L  <_  M
)  ->  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  (
# `  B ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B ) )  =  ( ( A substr  <. M ,  L >. ) concat 
( B substr  <. 0 ,  ( ( L  +  ( # `  B ) )  -  L )
>. ) ) )
4514, 31, 442if2 3980 . . 3  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  (
# `  B ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B ) )  =  if ( ( L  +  ( # `  B ) )  <_  L ,  ( A substr  <. M ,  ( L  +  ( # `  B
) ) >. ) ,  if ( L  <_  M ,  ( B substr  <.
( M  -  L
) ,  ( ( L  +  ( # `  B ) )  -  L ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat 
( B substr  <. 0 ,  ( ( L  +  ( # `  B ) )  -  L )
>. ) ) ) ) )
4613, 45eqtr4d 2504 . 2  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... ( L  +  ( # `  B
) ) ) )  ->  ( ( A concat  B ) substr  <. M , 
( L  +  (
# `  B )
) >. )  =  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  ( # `  B
) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B )
) )
4746ex 434 1  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( M  e.  ( 0 ... ( L  +  ( # `  B
) ) )  -> 
( ( A concat  B
) substr  <. M ,  ( L  +  ( # `  B ) ) >.
)  =  if ( L  <_  M , 
( B substr  <. ( M  -  L ) ,  ( # `  B
) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  B )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ifcif 3932   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481    + caddc 9484    <_ cle 9618    - cmin 9794   NN0cn0 10784   ZZ>=cuz 11071   ...cfz 11661   #chash 12360  Word cword 12487   concat cconcat 12489   substr csubstr 12491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-concat 12497  df-substr 12499
This theorem is referenced by: (None)
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