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Mirrors > Home > MPE Home > Th. List > pfxfv0 | Structured version Visualization version GIF version |
Description: The first symbol of a prefix is the first symbol of the word. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.) |
Ref | Expression |
---|---|
pfxfv0 | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝐿)‘0) = (𝑊‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(♯‘𝑊))) → 𝑊 ∈ Word 𝑉) | |
2 | fz1ssfz0 12998 | . . . 4 ⊢ (1...(♯‘𝑊)) ⊆ (0...(♯‘𝑊)) | |
3 | 2 | sseli 3911 | . . 3 ⊢ (𝐿 ∈ (1...(♯‘𝑊)) → 𝐿 ∈ (0...(♯‘𝑊))) |
4 | 3 | adantl 485 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(♯‘𝑊))) → 𝐿 ∈ (0...(♯‘𝑊))) |
5 | elfznn 12931 | . . . 4 ⊢ (𝐿 ∈ (1...(♯‘𝑊)) → 𝐿 ∈ ℕ) | |
6 | 5 | adantl 485 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(♯‘𝑊))) → 𝐿 ∈ ℕ) |
7 | lbfzo0 13072 | . . 3 ⊢ (0 ∈ (0..^𝐿) ↔ 𝐿 ∈ ℕ) | |
8 | 6, 7 | sylibr 237 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(♯‘𝑊))) → 0 ∈ (0..^𝐿)) |
9 | pfxfv 14035 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 0 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘0) = (𝑊‘0)) | |
10 | 1, 4, 8, 9 | syl3anc 1368 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝐿)‘0) = (𝑊‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 ℕcn 11625 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 prefix cpfx 14023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-substr 13994 df-pfx 14024 |
This theorem is referenced by: wwlksnredwwlkn0 27682 wwlksnextproplem1 27695 clwwlkinwwlk 27825 clwwlkf 27832 clwwlkvbij 27898 2clwwlklem 28128 clwwlknonclwlknonf1o 28147 numclwlk2lem2f 28162 cycpmco2lem4 30821 cycpmco2lem5 30822 |
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