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Theorem swrdlen 11424
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 1031 . . . . . . 7 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> S e. Word A )
2 elfzoelz 10554 . . . . . . . . 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 10428 . . . . . . . . . 10 |- ( F e. ( 0 ... L ) -> F e. ZZ )
543ad2ant2 1050 . . . . . . . . 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 9772 . . . . . . 7 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> ( x + F ) e. ZZ )
8 fvexg 5714 . . . . . . 7 |- ( ( S e. Word A /\ ( x + F ) e. ZZ ) -> ( S ` ( x + F ) ) e. _V )
91, 7, 8syl2anc 415 . . . . . 6 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> ( S ` ( x + F ) ) e. _V )
109ralrimiva 2623 . . . . 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 2238 . . . . . 6 |- ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) = ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) )
1211fnmpt 5510 . . . . 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 11423 . . . . 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 5471 . . . 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 9655 . . . 4 |- 0 e. ZZ
18 elfzelz 10428 . . . . . 6 |- ( L e. ( 0 ... (♯ ` S ) ) -> L e. ZZ )
19183ad2ant3 1051 . . . . 5 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> L e. ZZ )
2019, 5zsubcld 9773 . . . 4 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( L - F ) e. ZZ )
21 fzofig 10869 . . . 4 |- ( ( 0 e. ZZ /\ ( L - F ) e. ZZ ) -> ( 0..^ ( L - F ) ) e. Fin )
2217, 20, 21sylancr 418 . . 3 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( 0..^ ( L - F ) ) e. Fin )
23 fihashfn 11240 . . 3 |- ( ( ( S substr <. F , L >. ) Fn ( 0..^ ( L - F ) ) /\ ( 0..^ ( L - F ) ) e. Fin ) -> (♯ ` ( S substr <. F , L >. ) ) = (♯ ` ( 0..^ ( L - F ) ) ) )
2416, 22, 23syl2anc 415 . 2 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. F , L >. ) ) = (♯ ` ( 0..^ ( L - F ) ) ) )
25 fznn0sub 10463 . . . 4 |- ( F e. ( 0 ... L ) -> ( L - F ) e. NN0 )
26253ad2ant2 1050 . . 3 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( L - F ) e. NN0 )
27 hashfzo0 11264 . . 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 2271 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   <.cop 3712   |-> cmpt 4192   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Fincfn 7022   0cc0 8179   + caddc 8182   - cmin 8497   NN0cn0 9563   ZZcz 9644   ...cfz 10411  ..^cfzo 10549  ♯chash 11214  Word cword 11304   substr csubstr 11417
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-substr 11418
This theorem is used by:  swrdf  11427  swrdrlen  11433  swrdlen2  11434  swrds1  11440  ccatswrd  11442  swrdccat2  11443  ccatpfx  11473  swrdswrd  11477  pfxccatin12lem2  11503  pfxccatin12  11505
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