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Theorem swrdlen 11344
Description: Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
swrdlen |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. F , L >. ) ) = ( L - F ) )
Proof of Theorem swrdlen
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl1 1027 . . . . . . 7 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> S e. Word A )
2 elfzoelz 10481 . . . . . . . . 9 |- ( x e. ( 0..^ ( L - F ) ) -> x e. ZZ )
32adantl 277 . . . . . . . 8 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> x e. ZZ )
4 elfzelz 10359 . . . . . . . . . 10 |- ( F e. ( 0 ... L ) -> F e. ZZ )
543ad2ant2 1046 . . . . . . . . 9 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> F e. ZZ )
65adantr 276 . . . . . . . 8 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> F e. ZZ )
73, 6zaddcld 9704 . . . . . . 7 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> ( x + F ) e. ZZ )
8 fvexg 5689 . . . . . . 7 |- ( ( S e. Word A /\ ( x + F ) e. ZZ ) -> ( S ` ( x + F ) ) e. _V )
91, 7, 8syl2anc 411 . . . . . 6 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> ( S ` ( x + F ) ) e. _V )
109ralrimiva 2615 . . . . 5 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> A. x e. ( 0..^ ( L - F ) ) ( S ` ( x + F ) ) e. _V )
11 eqid 2232 . . . . . 6 |- ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) = ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) )
1211fnmpt 5485 . . . . 5 |- ( A. x e. ( 0..^ ( L - F ) ) ( S ` ( x + F ) ) e. _V -> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) Fn ( 0..^ ( L - F ) ) )
1310, 12syl 14 . . . 4 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) Fn ( 0..^ ( L - F ) ) )
14 swrdval2 11343 . . . . 5 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( S substr <. F , L >. ) = ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) )
1514fneq1d 5446 . . . 4 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( ( S substr <. F , L >. ) Fn ( 0..^ ( L - F ) ) <-> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) Fn ( 0..^ ( L - F ) ) ) )
1613, 15mpbird 167 . . 3 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( S substr <. F , L >. ) Fn ( 0..^ ( L - F ) ) )
17 0z 9588 . . . 4 |- 0 e. ZZ
18 elfzelz 10359 . . . . . 6 |- ( L e. ( 0 ... (♯ ` S ) ) -> L e. ZZ )
19183ad2ant3 1047 . . . . 5 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> L e. ZZ )
2019, 5zsubcld 9705 . . . 4 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( L - F ) e. ZZ )
21 fzofig 10794 . . . 4 |- ( ( 0 e. ZZ /\ ( L - F ) e. ZZ ) -> ( 0..^ ( L - F ) ) e. Fin )
2217, 20, 21sylancr 414 . . 3 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( 0..^ ( L - F ) ) e. Fin )
23 fihashfn 11164 . . 3 |- ( ( ( S substr <. F , L >. ) Fn ( 0..^ ( L - F ) ) /\ ( 0..^ ( L - F ) ) e. Fin ) -> (♯ ` ( S substr <. F , L >. ) ) = (♯ ` ( 0..^ ( L - F ) ) ) )
2416, 22, 23syl2anc 411 . 2 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. F , L >. ) ) = (♯ ` ( 0..^ ( L - F ) ) ) )
25 fznn0sub 10391 . . . 4 |- ( F e. ( 0 ... L ) -> ( L - F ) e. NN0 )
26253ad2ant2 1046 . . 3 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( L - F ) e. NN0 )
27 hashfzo0 11188 . . 3 |- ( ( L - F ) e. NN0 -> (♯ ` ( 0..^ ( L - F ) ) ) = ( L - F ) )
2826, 27syl 14 . 2 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( 0..^ ( L - F ) ) ) = ( L - F ) )
2924, 28eqtrd 2265 1 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. F , L >. ) ) = ( L - F ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   <.cop 3692   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Fincfn 6975   0cc0 8127   + caddc 8130   - cmin 8444   NN0cn0 9496   ZZcz 9577   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   substr csubstr 11337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-substr 11338
This theorem is referenced by:  swrdf  11347  swrdrlen  11353  swrdlen2  11354  swrds1  11360  ccatswrd  11362  swrdccat2  11363  ccatpfx  11393  swrdswrd  11397  pfxccatin12lem2  11423  pfxccatin12  11425
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