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Theorem swrdccat2 12640
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 12552 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S concat  T )  e. Word  B )
2 swrdcl 12603 . . . 4  |-  ( ( S concat  T )  e. Word  B  ->  ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
)  e. Word  B )
3 wrdf 12513 . . . 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 )
4 ffn 5729 . . . 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 )
) >. ) ) ) )
51, 2, 3, 44syl 21 . . 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
) ) >. )
) ) )
6 lencl 12522 . . . . . . . . . 10  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
76adantr 465 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  NN0 )
8 nn0uz 11112 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
97, 8syl6eleq 2565 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
107nn0zd 10960 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ZZ )
11 uzid 11092 . . . . . . . . . 10  |-  ( (
# `  S )  e.  ZZ  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
1210, 11syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
13 lencl 12522 . . . . . . . . . 10  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
1413adantl 466 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  NN0 )
15 uzaddcl 11133 . . . . . . . . 9  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) )  /\  ( # `  T )  e.  NN0 )  ->  ( ( # `  S )  +  (
# `  T )
)  e.  ( ZZ>= `  ( # `  S ) ) )
1612, 14, 15syl2anc 661 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) ) )
17 elfzuzb 11678 . . . . . . . 8  |-  ( (
# `  S )  e.  ( 0 ... (
( # `  S )  +  ( # `  T
) ) )  <->  ( ( # `
 S )  e.  ( ZZ>= `  0 )  /\  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) ) ) )
189, 16, 17sylanbrc 664 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( 0 ... ( ( # `  S )  +  (
# `  T )
) ) )
197, 14nn0addcld 10852 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e. 
NN0 )
2019, 8syl6eleq 2565 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  0 )
)
2119nn0zd 10960 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ZZ )
22 uzid 11092 . . . . . . . . . 10  |-  ( ( ( # `  S
)  +  ( # `  T ) )  e.  ZZ  ->  ( ( # `
 S )  +  ( # `  T
) )  e.  (
ZZ>= `  ( ( # `  S )  +  (
# `  T )
) ) )
2321, 22syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( ( # `
 S )  +  ( # `  T
) ) ) )
24 elfzuzb 11678 . . . . . . . . 9  |-  ( ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... (
( # `  S )  +  ( # `  T
) ) )  <->  ( (
( # `  S )  +  ( # `  T
) )  e.  (
ZZ>= `  0 )  /\  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( ( # `
 S )  +  ( # `  T
) ) ) ) )
2520, 23, 24sylanbrc 664 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... (
( # `  S )  +  ( # `  T
) ) ) )
26 ccatlen 12553 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  ( S concat  T ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
2726oveq2d 6298 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0 ... ( # `
 ( S concat  T
) ) )  =  ( 0 ... (
( # `  S )  +  ( # `  T
) ) ) )
2825, 27eleqtrrd 2558 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... ( # `
 ( S concat  T
) ) ) )
29 swrdlen 12607 . . . . . . 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 ) ) )
301, 18, 28, 29syl3anc 1228 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) )  =  ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) )
317nn0cnd 10850 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  CC )
3214nn0cnd 10850 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  CC )
3331, 32pncan2d 9928 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) )  =  ( # `  T
) )
3430, 33eqtrd 2508 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) )  =  ( # `  T
) )
3534oveq2d 6298 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( # `  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )
) )  =  ( 0..^ ( # `  T
) ) )
3635fneq2d 5670 . . 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 ) ) ) )
375, 36mpbid 210 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  Fn  ( 0..^ ( # `  T ) ) )
38 wrdf 12513 . . . 4  |-  ( T  e. Word  B  ->  T : ( 0..^ (
# `  T )
) --> B )
3938adantl 466 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  T : ( 0..^ ( # `  T
) ) --> B )
40 ffn 5729 . . 3  |-  ( T : ( 0..^ (
# `  T )
) --> B  ->  T  Fn  ( 0..^ ( # `  T ) ) )
4139, 40syl 16 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  T  Fn  ( 0..^ ( # `  T
) ) )
421adantr 465 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( S concat  T )  e. Word  B )
4318adantr 465 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( # `  S )  e.  ( 0 ... ( ( # `  S
)  +  ( # `  T ) ) ) )
4428adantr 465 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... ( # `
 ( S concat  T
) ) ) )
4533oveq2d 6298 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( ( ( # `  S
)  +  ( # `  T ) )  -  ( # `  S ) ) )  =  ( 0..^ ( # `  T
) ) )
4645eleq2d 2537 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( k  e.  ( 0..^ ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) ) )  <->  k  e.  ( 0..^ ( # `  T
) ) ) )
4746biimpar 485 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
k  e.  ( 0..^ ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) ) )
48 swrdfv 12608 . . . 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 ) ) ) )
4942, 43, 44, 47, 48syl31anc 1231 . . 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 ) ) ) )
50 ccatval3 12556 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  k  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
k  +  ( # `  S ) ) )  =  ( T `  k ) )
51503expa 1196 . . 3  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( ( S concat  T
) `  ( k  +  ( # `  S
) ) )  =  ( T `  k
) )
5249, 51eqtrd 2508 . 2  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  T
) ) )  -> 
( ( ( S concat  T ) substr  <. ( # `  S ) ,  ( ( # `  S
)  +  ( # `  T ) ) >.
) `  k )  =  ( T `  k ) )
5337, 41, 52eqfnfvd 5976 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )  =  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   0cc0 9488    + caddc 9491    - cmin 9801   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668  ..^cfzo 11788   #chash 12367  Word cword 12494   concat cconcat 12496   substr csubstr 12498
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 12368  df-word 12502  df-concat 12504  df-substr 12506
This theorem is referenced by:  ccatopth  12652
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