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Theorem pfxswrd 11423
Description: A prefix of a subword is a subword. (Contributed by AV, 2-Apr-2018.) (Revised by AV, 8-May-2020.)
Assertion
Ref Expression
pfxswrd |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( L e. ( 0 ... ( N - M ) ) -> ( ( W substr <. M , N >. ) prefix L ) = ( W substr <. M , ( M + L ) >. ) ) )
Proof of Theorem pfxswrd
StepHypRef Expression
1 simp1 1024 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> W e. Word V )
2 elfzelz 10378 . . . . . 6 |- ( M e. ( 0 ... N ) -> M e. ZZ )
323ad2ant3 1047 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> M e. ZZ )
4 elfzel2 10376 . . . . . 6 |- ( M e. ( 0 ... N ) -> N e. ZZ )
543ad2ant3 1047 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> N e. ZZ )
6 swrdclg 11367 . . . . 5 |- ( ( W e. Word V /\ M e. ZZ /\ N e. ZZ ) -> ( W substr <. M , N >. ) e. Word V )
71, 3, 5, 6syl3anc 1274 . . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( W substr <. M , N >. ) e. Word V )
8 elfznn0 10470 . . . 4 |- ( L e. ( 0 ... ( N - M ) ) -> L e. NN0 )
9 pfxval 11391 . . . 4 |- ( ( ( W substr <. M , N >. ) e. Word V /\ L e. NN0 ) -> ( ( W substr <. M , N >. ) prefix L ) = ( ( W substr <. M , N >. ) substr <. 0 , L >. ) )
107, 8, 9syl2an 289 . . 3 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> ( ( W substr <. M , N >. ) prefix L ) = ( ( W substr <. M , N >. ) substr <. 0 , L >. ) )
11 fznn0sub 10412 . . . . . . 7 |- ( M e. ( 0 ... N ) -> ( N - M ) e. NN0 )
12113ad2ant3 1047 . . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( N - M ) e. NN0 )
13 0elfz 10474 . . . . . 6 |- ( ( N - M ) e. NN0 -> 0 e. ( 0 ... ( N - M ) ) )
1412, 13syl 14 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> 0 e. ( 0 ... ( N - M ) ) )
1514anim1i 340 . . . 4 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> ( 0 e. ( 0 ... ( N - M ) ) /\ L e. ( 0 ... ( N - M ) ) ) )
16 swrdswrd 11422 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( ( 0 e. ( 0 ... ( N - M ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> ( ( W substr <. M , N >. ) substr <. 0 , L >. ) = ( W substr <. ( M + 0 ) , ( M + L ) >. ) ) )
1716imp 124 . . . 4 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ ( 0 e. ( 0 ... ( N - M ) ) /\ L e. ( 0 ... ( N - M ) ) ) ) -> ( ( W substr <. M , N >. ) substr <. 0 , L >. ) = ( W substr <. ( M + 0 ) , ( M + L ) >. ) )
1815, 17syldan 282 . . 3 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> ( ( W substr <. M , N >. ) substr <. 0 , L >. ) = ( W substr <. ( M + 0 ) , ( M + L ) >. ) )
19 elfznn0 10470 . . . . . . . 8 |- ( M e. ( 0 ... N ) -> M e. NN0 )
20 nn0cn 9523 . . . . . . . . 9 |- ( M e. NN0 -> M e. CC )
2120addridd 8438 . . . . . . . 8 |- ( M e. NN0 -> ( M + 0 ) = M )
2219, 21syl 14 . . . . . . 7 |- ( M e. ( 0 ... N ) -> ( M + 0 ) = M )
23223ad2ant3 1047 . . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( M + 0 ) = M )
2423adantr 276 . . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> ( M + 0 ) = M )
2524opeq1d 3894 . . . 4 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> <. ( M + 0 ) , ( M + L ) >. = <. M , ( M + L ) >. )
2625oveq2d 6074 . . 3 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> ( W substr <. ( M + 0 ) , ( M + L ) >. ) = ( W substr <. M , ( M + L ) >. ) )
2710, 18, 263eqtrd 2271 . 2 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) /\ L e. ( 0 ... ( N - M ) ) ) -> ( ( W substr <. M , N >. ) prefix L ) = ( W substr <. M , ( M + L ) >. ) )
2827ex 115 1 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( L e. ( 0 ... ( N - M ) ) -> ( ( W substr <. M , N >. ) prefix L ) = ( W substr <. M , ( M + L ) >. ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   <.cop 3697   ` cfv 5357  (class class class)co 6058   0cc0 8143   + caddc 8146   - cmin 8460   NN0cn0 9513   ZZcz 9594   ...cfz 10361  ♯chash 11163  Word cword 11249   substr csubstr 11362   prefix cpfx 11389
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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259
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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-substr 11363  df-pfx 11390
This theorem is referenced by:  pfxpfx  11425
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