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Theorem pfxswrd 11165
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 1000 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> W e. Word V )
2 elfzelz 10154 . . . . . 6 |- ( M e. ( 0 ... N ) -> M e. ZZ )
323ad2ant3 1023 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> M e. ZZ )
4 elfzel2 10152 . . . . . 6 |- ( M e. ( 0 ... N ) -> N e. ZZ )
543ad2ant3 1023 . . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> N e. ZZ )
6 swrdclg 11111 . . . . 5 |- ( ( W e. Word V /\ M e. ZZ /\ N e. ZZ ) -> ( W substr <. M , N >. ) e. Word V )
71, 3, 5, 6syl3anc 1250 . . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( W substr <. M , N >. ) e. Word V )
8 elfznn0 10243 . . . 4 |- ( L e. ( 0 ... ( N - M ) ) -> L e. NN0 )
9 pfxval 11135 . . . 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 10186 . . . . . . 7 |- ( M e. ( 0 ... N ) -> ( N - M ) e. NN0 )
12113ad2ant3 1023 . . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ M e. ( 0 ... N ) ) -> ( N - M ) e. NN0 )
13 0elfz 10247 . . . . . 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 11164 . . . . 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 10243 . . . . . . . 8 |- ( M e. ( 0 ... N ) -> M e. NN0 )
20 nn0cn 9312 . . . . . . . . 9 |- ( M e. NN0 -> M e. CC )
2120addridd 8228 . . . . . . . 8 |- ( M e. NN0 -> ( M + 0 ) = M )
2219, 21syl 14 . . . . . . 7 |- ( M e. ( 0 ... N ) -> ( M + 0 ) = M )
23223ad2ant3 1023 . . . . . 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 3827 . . . 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 5967 . . 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 2243 . 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 981   = wceq 1373   e. wcel 2177   <.cop 3637   ` cfv 5276  (class class class)co 5951   0cc0 7932   + caddc 7935   - cmin 8250   NN0cn0 9302   ZZcz 9379   ...cfz 10137  ♯chash 10927  Word cword 11001   substr csubstr 11106   prefix cpfx 11133
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 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048
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 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-1o 6509  df-er 6627  df-en 6835  df-dom 6836  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-inn 9044  df-n0 9303  df-z 9380  df-uz 9656  df-fz 10138  df-fzo 10272  df-ihash 10928  df-word 11002  df-substr 11107  df-pfx 11134
This theorem is referenced by:  pfxpfx  11167
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