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Theorem swrdlen 11138
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 1003 . . . . . . 7 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> S e. Word A )
2 elfzoelz 10299 . . . . . . . . 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 10177 . . . . . . . . . 10 |- ( F e. ( 0 ... L ) -> F e. ZZ )
543ad2ant2 1022 . . . . . . . . 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 9529 . . . . . . 7 |- ( ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( L - F ) ) ) -> ( x + F ) e. ZZ )
8 fvexg 5613 . . . . . . 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 2580 . . . . 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 2206 . . . . . 6 |- ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) = ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) )
1211fnmpt 5417 . . . . 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 11137 . . . . 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 5378 . . . 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 9413 . . . 4 |- 0 e. ZZ
18 elfzelz 10177 . . . . . 6 |- ( L e. ( 0 ... (♯ ` S ) ) -> L e. ZZ )
19183ad2ant3 1023 . . . . 5 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> L e. ZZ )
2019, 5zsubcld 9530 . . . 4 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( L - F ) e. ZZ )
21 fzofig 10609 . . . 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 10977 . . 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 10209 . . . 4 |- ( F e. ( 0 ... L ) -> ( L - F ) e. NN0 )
26253ad2ant2 1022 . . 3 |- ( ( S e. Word A /\ F e. ( 0 ... L ) /\ L e. ( 0 ... (♯ ` S ) ) ) -> ( L - F ) e. NN0 )
27 hashfzo0 11000 . . 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 2239 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 981   = wceq 1373   e. wcel 2177   A.wral 2485   _Vcvv 2773   <.cop 3641   |-> cmpt 4116   Fn wfn 5280   ` cfv 5285  (class class class)co 5962   Fincfn 6845   0cc0 7955   + caddc 7958   - cmin 8273   NN0cn0 9325   ZZcz 9402   ...cfz 10160  ..^cfzo 10294  ♯chash 10952  Word cword 11026   substr csubstr 11131
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-1o 6520  df-er 6638  df-en 6846  df-dom 6847  df-fin 6848  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-fz 10161  df-fzo 10295  df-ihash 10953  df-word 11027  df-substr 11132
This theorem is referenced by:  swrdf  11141  swrdrlen  11147  swrdlen2  11148  swrds1  11154  ccatswrd  11156  swrdccat2  11157  ccatpfx  11187  swrdswrd  11191
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