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Theorem swrdccat2 11730
Description: Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
swrdccat2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  =  T )

Proof of Theorem swrdccat2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 ccatcl 11698 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S concat  T )  e. Word  B )
2 swrdcl 11721 . . . . 5  |-  ( ( S concat  T )  e. Word  B  ->  ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
)  e. Word  B )
31, 2syl 16 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  e. Word  B )
4 wrdf 11688 . . . 4  |-  ( ( ( S concat  T ) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  e. Word  B  ->  ( ( S concat  T ) substr  <. ( # `
 S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
) : ( 0..^ ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) ) ) --> B )
5 ffn 5550 . . . 4  |-  ( ( ( S concat  T ) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. ) : ( 0..^ (
# `  ( ( S concat  T ) substr  <. ( # `
 S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
) ) ) --> B  ->  ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
)  Fn  ( 0..^ ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) ) ) )
63, 4, 53syl 19 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  Fn  ( 0..^ ( # `  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )
) ) )
7 lencl 11690 . . . . . . . . . 10  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
87adantr 452 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  NN0 )
9 nn0uz 10476 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
108, 9syl6eleq 2494 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
118nn0zd 10329 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ZZ )
12 uzid 10456 . . . . . . . . . 10  |-  ( (
# `  S )  e.  ZZ  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
1311, 12syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
14 lencl 11690 . . . . . . . . . 10  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
1514adantl 453 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  NN0 )
16 uzaddcl 10489 . . . . . . . . 9  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) )  /\  ( # `  T )  e.  NN0 )  ->  ( ( # `  S )  +  (
# `  T )
)  e.  ( ZZ>= `  ( # `  S ) ) )
1713, 15, 16syl2anc 643 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) ) )
18 elfzuzb 11009 . . . . . . . 8  |-  ( (
# `  S )  e.  ( 0 ... (
( # `  S )  +  ( # `  T
) ) )  <->  ( ( # `
 S )  e.  ( ZZ>= `  0 )  /\  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) ) ) )
1910, 17, 18sylanbrc 646 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( 0 ... ( ( # `  S )  +  (
# `  T )
) ) )
208, 15nn0addcld 10234 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e. 
NN0 )
2120, 9syl6eleq 2494 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  0 )
)
2220nn0zd 10329 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ZZ )
23 uzid 10456 . . . . . . . . . 10  |-  ( ( ( # `  S
)  +  ( # `  T ) )  e.  ZZ  ->  ( ( # `
 S )  +  ( # `  T
) )  e.  (
ZZ>= `  ( ( # `  S )  +  (
# `  T )
) ) )
2422, 23syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( ( # `
 S )  +  ( # `  T
) ) ) )
25 elfzuzb 11009 . . . . . . . . 9  |-  ( ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... (
( # `  S )  +  ( # `  T
) ) )  <->  ( (
( # `  S )  +  ( # `  T
) )  e.  (
ZZ>= `  0 )  /\  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( ( # `
 S )  +  ( # `  T
) ) ) ) )
2621, 24, 25sylanbrc 646 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... (
( # `  S )  +  ( # `  T
) ) ) )
27 ccatlen 11699 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  ( S concat  T ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
2827oveq2d 6056 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0 ... ( # `
 ( S concat  T
) ) )  =  ( 0 ... (
( # `  S )  +  ( # `  T
) ) ) )
2926, 28eleqtrrd 2481 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... ( # `
 ( S concat  T
) ) ) )
30 swrdlen 11725 . . . . . . 7  |-  ( ( ( S concat  T )  e. Word  B  /\  ( # `
 S )  e.  ( 0 ... (
( # `  S )  +  ( # `  T
) ) )  /\  ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... ( # `
 ( S concat  T
) ) ) )  ->  ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) )  =  ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) )
311, 19, 29, 30syl3anc 1184 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) )  =  ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) )
328nn0cnd 10232 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  CC )
3315nn0cnd 10232 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  CC )
3432, 33pncan2d 9369 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) )  =  ( # `  T
) )
3531, 34eqtrd 2436 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) )  =  ( # `  T
) )
3635oveq2d 6056 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( # `  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )
) )  =  ( 0..^ ( # `  T
) ) )
3736fneq2d 5496 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
)  Fn  ( 0..^ ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) ) )  <-> 
( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  Fn  ( 0..^ ( # `  T ) ) ) )
386, 37mpbid 202 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  Fn  ( 0..^ ( # `  T ) ) )
39 wrdf 11688 . . . 4  |-  ( T  e. Word  B  ->  T : ( 0..^ (
# `  T )
) --> B )
4039adantl 453 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  T : ( 0..^ ( # `  T
) ) --> B )
41 ffn 5550 . . 3  |-  ( T : ( 0..^ (
# `  T )
) --> B  ->  T  Fn  ( 0..^ ( # `  T ) ) )
4240, 41syl 16 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  T  Fn  ( 0..^ ( # `  T
) ) )
431adantr 452 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( S concat  T )  e. Word  B )
4419adantr 452 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( # `  S )  e.  ( 0 ... ( ( # `  S
)  +  ( # `  T ) ) ) )
4529adantr 452 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... ( # `
 ( S concat  T
) ) ) )
4634oveq2d 6056 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( ( ( # `  S
)  +  ( # `  T ) )  -  ( # `  S ) ) )  =  ( 0..^ ( # `  T
) ) )
4746eleq2d 2471 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( k  e.  ( 0..^ ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) ) )  <->  k  e.  ( 0..^ ( # `  T
) ) ) )
4847biimpar 472 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
k  e.  ( 0..^ ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) ) )
49 swrdfv 11726 . . . 4  |-  ( ( ( ( S concat  T
)  e. Word  B  /\  ( # `  S )  e.  ( 0 ... ( ( # `  S
)  +  ( # `  T ) ) )  /\  ( ( # `  S )  +  (
# `  T )
)  e.  ( 0 ... ( # `  ( S concat  T ) ) ) )  /\  k  e.  ( 0..^ ( ( ( # `  S
)  +  ( # `  T ) )  -  ( # `  S ) ) ) )  -> 
( ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
) `  k )  =  ( ( S concat  T ) `  (
k  +  ( # `  S ) ) ) )
5043, 44, 45, 48, 49syl31anc 1187 . . 3  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
) `  k )  =  ( ( S concat  T ) `  (
k  +  ( # `  S ) ) ) )
51 ccatval3 11702 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  k  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
k  +  ( # `  S ) ) )  =  ( T `  k ) )
52513expa 1153 . . 3  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( ( S concat  T
) `  ( k  +  ( # `  S
) ) )  =  ( T `  k
) )
5350, 52eqtrd 2436 . 2  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
) `  k )  =  ( T `  k ) )
5438, 42, 53eqfnfvd 5789 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3777    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946    + caddc 8949    - cmin 9247   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   concat cconcat 11673   substr csubstr 11675
This theorem is referenced by:  ccatopth  11731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-substr 11681
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