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Theorem swrdccat1 12676
Description: Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
swrdccat1  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T
) substr  <. 0 ,  (
# `  S ) >. )  =  S )

Proof of Theorem swrdccat1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 ccatcl 12585 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S ++  T )  e. Word  B )
2 lencl 12552 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
32adantr 463 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  NN0 )
4 nn0uz 11116 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
53, 4syl6eleq 2552 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
6 ccatlen 12586 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  ( S ++  T ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
73nn0zd 10963 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ZZ )
8 uzid 11096 . . . . . . 7  |-  ( (
# `  S )  e.  ZZ  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
97, 8syl 16 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
10 lencl 12552 . . . . . . 7  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
1110adantl 464 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  NN0 )
12 uzaddcl 11138 . . . . . 6  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) )  /\  ( # `  T )  e.  NN0 )  ->  ( ( # `  S )  +  (
# `  T )
)  e.  ( ZZ>= `  ( # `  S ) ) )
139, 11, 12syl2anc 659 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) ) )
146, 13eqeltrd 2542 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  ( S ++  T ) )  e.  ( ZZ>= `  ( # `  S
) ) )
15 elfzuzb 11685 . . . 4  |-  ( (
# `  S )  e.  ( 0 ... ( # `
 ( S ++  T
) ) )  <->  ( ( # `
 S )  e.  ( ZZ>= `  0 )  /\  ( # `  ( S ++  T ) )  e.  ( ZZ>= `  ( # `  S
) ) ) )
165, 14, 15sylanbrc 662 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( 0 ... ( # `  ( S ++  T ) ) ) )
17 swrd0val 12640 . . 3  |-  ( ( ( S ++  T )  e. Word  B  /\  ( # `
 S )  e.  ( 0 ... ( # `
 ( S ++  T
) ) ) )  ->  ( ( S ++  T ) substr  <. 0 ,  ( # `  S
) >. )  =  ( ( S ++  T )  |`  ( 0..^ ( # `  S ) ) ) )
181, 16, 17syl2anc 659 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T
) substr  <. 0 ,  (
# `  S ) >. )  =  ( ( S ++  T )  |`  ( 0..^ ( # `  S
) ) ) )
19 wrdf 12541 . . . . 5  |-  ( ( S ++  T )  e. Word  B  ->  ( S ++  T
) : ( 0..^ ( # `  ( S ++  T ) ) ) --> B )
20 ffn 5713 . . . . 5  |-  ( ( S ++  T ) : ( 0..^ ( # `  ( S ++  T ) ) ) --> B  -> 
( S ++  T )  Fn  ( 0..^ (
# `  ( S ++  T ) ) ) )
211, 19, 203syl 20 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S ++  T )  Fn  ( 0..^ (
# `  ( S ++  T ) ) ) )
22 fzoss2 11830 . . . . 5  |-  ( (
# `  ( S ++  T ) )  e.  ( ZZ>= `  ( # `  S
) )  ->  (
0..^ ( # `  S
) )  C_  (
0..^ ( # `  ( S ++  T ) ) ) )
2314, 22syl 16 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( # `  S ) )  C_  ( 0..^ ( # `  ( S ++  T ) ) ) )
24 fnssres 5676 . . . 4  |-  ( ( ( S ++  T )  Fn  ( 0..^ (
# `  ( S ++  T ) ) )  /\  ( 0..^ (
# `  S )
)  C_  ( 0..^ ( # `  ( S ++  T ) ) ) )  ->  ( ( S ++  T )  |`  (
0..^ ( # `  S
) ) )  Fn  ( 0..^ ( # `  S ) ) )
2521, 23, 24syl2anc 659 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T
)  |`  ( 0..^ (
# `  S )
) )  Fn  (
0..^ ( # `  S
) ) )
26 wrdf 12541 . . . . 5  |-  ( S  e. Word  B  ->  S : ( 0..^ (
# `  S )
) --> B )
2726adantr 463 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S : ( 0..^ ( # `  S
) ) --> B )
28 ffn 5713 . . . 4  |-  ( S : ( 0..^ (
# `  S )
) --> B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
2927, 28syl 16 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S  Fn  ( 0..^ ( # `  S
) ) )
30 fvres 5862 . . . . 5  |-  ( k  e.  ( 0..^ (
# `  S )
)  ->  ( (
( S ++  T )  |`  ( 0..^ ( # `  S ) ) ) `
 k )  =  ( ( S ++  T
) `  k )
)
3130adantl 464 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( ( S ++  T )  |`  (
0..^ ( # `  S
) ) ) `  k )  =  ( ( S ++  T ) `
 k ) )
32 ccatval1 12587 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  k  e.  ( 0..^ ( # `  S ) ) )  ->  ( ( S ++  T ) `  k
)  =  ( S `
 k ) )
33323expa 1194 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( S ++  T
) `  k )  =  ( S `  k ) )
3431, 33eqtrd 2495 . . 3  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( ( S ++  T )  |`  (
0..^ ( # `  S
) ) ) `  k )  =  ( S `  k ) )
3525, 29, 34eqfnfvd 5960 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T
)  |`  ( 0..^ (
# `  S )
) )  =  S )
3618, 35eqtrd 2495 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T
) substr  <. 0 ,  (
# `  S ) >. )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   <.cop 4022    |` cres 4990    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   0cc0 9481    + caddc 9484   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675  ..^cfzo 11799   #chash 12390  Word cword 12521   ++ cconcat 12523   substr csubstr 12525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-concat 12531  df-substr 12533
This theorem is referenced by:  ccatopth  12689  reuccats1  12700  wwlknextbi  24930  wwlkextsur  24936  clwwlkfo  25002
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