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Mirrors > Home > MPE Home > Th. List > pfxlen | Structured version Visualization version GIF version |
Description: Length of a prefix. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by AV, 2-May-2020.) |
Ref | Expression |
---|---|
pfxlen | ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝐿)) = 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pfxfn 14631 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) Fn (0..^𝐿)) | |
2 | hashfn 14335 | . . 3 ⊢ ((𝑆 prefix 𝐿) Fn (0..^𝐿) → (♯‘(𝑆 prefix 𝐿)) = (♯‘(0..^𝐿))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝐿)) = (♯‘(0..^𝐿))) |
4 | elfznn0 13594 | . . . 4 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) → 𝐿 ∈ ℕ0) | |
5 | 4 | adantl 483 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → 𝐿 ∈ ℕ0) |
6 | hashfzo0 14390 | . . 3 ⊢ (𝐿 ∈ ℕ0 → (♯‘(0..^𝐿)) = 𝐿) | |
7 | 5, 6 | syl 17 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘(0..^𝐿)) = 𝐿) |
8 | 3, 7 | eqtrd 2773 | 1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝐿)) = 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Fn wfn 6539 ‘cfv 6544 (class class class)co 7409 0cc0 11110 ℕ0cn0 12472 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 Word cword 14464 prefix cpfx 14620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-substr 14591 df-pfx 14621 |
This theorem is referenced by: addlenrevpfx 14640 addlenpfx 14641 pfxfvlsw 14645 pfxeq 14646 ccatpfx 14651 wrdind 14672 wrd2ind 14673 pfxccatin12 14683 spllen 14704 splfv1 14705 splfv2a 14706 splval2 14707 repswpfx 14735 cshwlen 14749 cshwidxmod 14753 pfx2 14898 efgsres 19606 efgredleme 19611 efgredlemd 19612 efgredlemc 19613 efgredlem 19615 efgcpbllemb 19623 wlkres 28927 trlreslem 28956 wwlksm1edg 29135 wwlksnred 29146 wwlksnextproplem3 29165 clwlkclwwlk 29255 clwwlkinwwlk 29293 clwwlkf 29300 wwlksubclwwlk 29311 clwlknf1oclwwlknlem1 29334 pfxlsw2ccat 32116 wrdt2ind 32117 splfv3 32122 cycpmco2lem2 32286 cycpmco2lem3 32287 cycpmco2lem4 32288 cycpmco2lem5 32289 cycpmco2lem6 32290 cycpmco2 32292 signstfveq0 33588 revpfxsfxrev 34106 swrdrevpfx 34107 pfxwlk 34114 swrdwlk 34117 |
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