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Theorem swrdccat2 11775
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 11743 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S concat  T )  e. Word  B )
2 swrdcl 11766 . . . . 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 11733 . . . 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 5591 . . . 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 11735 . . . . . . . . . 10  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
87adantr 452 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  NN0 )
9 nn0uz 10520 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
108, 9syl6eleq 2526 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
118nn0zd 10373 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ZZ )
12 uzid 10500 . . . . . . . . . 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 11735 . . . . . . . . . 10  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
1514adantl 453 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  NN0 )
16 uzaddcl 10533 . . . . . . . . 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 11053 . . . . . . . 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 10278 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e. 
NN0 )
2120, 9syl6eleq 2526 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  0 )
)
2220nn0zd 10373 . . . . . . . . . 10  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ZZ )
23 uzid 10500 . . . . . . . . . 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 11053 . . . . . . . . 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 11744 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  ( S concat  T ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
2827oveq2d 6097 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0 ... ( # `
 ( S concat  T
) ) )  =  ( 0 ... (
( # `  S )  +  ( # `  T
) ) ) )
2926, 28eleqtrrd 2513 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( 0 ... ( # `
 ( S concat  T
) ) ) )
30 swrdlen 11770 . . . . . . 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 10276 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  CC )
3315nn0cnd 10276 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  CC )
3432, 33pncan2d 9413 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) )  =  ( # `  T
) )
3531, 34eqtrd 2468 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  (
( S concat  T ) substr  <.
( # `  S ) ,  ( ( # `  S )  +  (
# `  T )
) >. ) )  =  ( # `  T
) )
3635oveq2d 6097 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( # `  ( ( S concat  T
) substr  <. ( # `  S
) ,  ( (
# `  S )  +  ( # `  T
) ) >. )
) )  =  ( 0..^ ( # `  T
) ) )
3736fneq2d 5537 . . 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 11733 . . . 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 5591 . . 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 6097 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( ( ( # `  S
)  +  ( # `  T ) )  -  ( # `  S ) ) )  =  ( 0..^ ( # `  T
) ) )
4746eleq2d 2503 . . . . 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 11771 . . . 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 11747 . . . 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 2468 . 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 5830 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 1652    e. wcel 1725   <.cop 3817    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990    + caddc 8993    - cmin 9291   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043  ..^cfzo 11135   #chash 11618  Word cword 11717   concat cconcat 11718   substr csubstr 11720
This theorem is referenced by:  ccatopth  11776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724  df-substr 11726
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