![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pfxswrd | Structured version Visualization version GIF version |
Description: A prefix of a subword is a subword. (Contributed by AV, 2-Apr-2018.) (Revised by AV, 8-May-2020.) |
Ref | Expression |
---|---|
pfxswrd | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝐿 ∈ (0...(𝑁 − 𝑀)) → ((𝑊 substr 〈𝑀, 𝑁〉) prefix 𝐿) = (𝑊 substr 〈𝑀, (𝑀 + 𝐿)〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovexd 6958 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 substr 〈𝑀, 𝑁〉) ∈ V) | |
2 | elfznn0 12756 | . . . 4 ⊢ (𝐿 ∈ (0...(𝑁 − 𝑀)) → 𝐿 ∈ ℕ0) | |
3 | pfxval 13788 | . . . 4 ⊢ (((𝑊 substr 〈𝑀, 𝑁〉) ∈ V ∧ 𝐿 ∈ ℕ0) → ((𝑊 substr 〈𝑀, 𝑁〉) prefix 𝐿) = ((𝑊 substr 〈𝑀, 𝑁〉) substr 〈0, 𝐿〉)) | |
4 | 1, 2, 3 | syl2an 589 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → ((𝑊 substr 〈𝑀, 𝑁〉) prefix 𝐿) = ((𝑊 substr 〈𝑀, 𝑁〉) substr 〈0, 𝐿〉)) |
5 | fznn0sub 12695 | . . . . . . 7 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ0) | |
6 | 5 | 3ad2ant3 1126 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑁 − 𝑀) ∈ ℕ0) |
7 | 0elfz 12760 | . . . . . 6 ⊢ ((𝑁 − 𝑀) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 𝑀))) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → 0 ∈ (0...(𝑁 − 𝑀))) |
9 | 8 | anim1i 608 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → (0 ∈ (0...(𝑁 − 𝑀)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀)))) |
10 | swrdswrd 13820 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → ((0 ∈ (0...(𝑁 − 𝑀)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → ((𝑊 substr 〈𝑀, 𝑁〉) substr 〈0, 𝐿〉) = (𝑊 substr 〈(𝑀 + 0), (𝑀 + 𝐿)〉))) | |
11 | 10 | imp 397 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ (0 ∈ (0...(𝑁 − 𝑀)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀)))) → ((𝑊 substr 〈𝑀, 𝑁〉) substr 〈0, 𝐿〉) = (𝑊 substr 〈(𝑀 + 0), (𝑀 + 𝐿)〉)) |
12 | 9, 11 | syldan 585 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → ((𝑊 substr 〈𝑀, 𝑁〉) substr 〈0, 𝐿〉) = (𝑊 substr 〈(𝑀 + 0), (𝑀 + 𝐿)〉)) |
13 | elfznn0 12756 | . . . . . . . 8 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0) | |
14 | nn0cn 11658 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ) | |
15 | 14 | addid1d 10578 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 0) = 𝑀) |
16 | 13, 15 | syl 17 | . . . . . . 7 ⊢ (𝑀 ∈ (0...𝑁) → (𝑀 + 0) = 𝑀) |
17 | 16 | 3ad2ant3 1126 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑀 + 0) = 𝑀) |
18 | 17 | adantr 474 | . . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 0) = 𝑀) |
19 | 18 | opeq1d 4644 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → 〈(𝑀 + 0), (𝑀 + 𝐿)〉 = 〈𝑀, (𝑀 + 𝐿)〉) |
20 | 19 | oveq2d 6940 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → (𝑊 substr 〈(𝑀 + 0), (𝑀 + 𝐿)〉) = (𝑊 substr 〈𝑀, (𝑀 + 𝐿)〉)) |
21 | 4, 12, 20 | 3eqtrd 2818 | . 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → ((𝑊 substr 〈𝑀, 𝑁〉) prefix 𝐿) = (𝑊 substr 〈𝑀, (𝑀 + 𝐿)〉)) |
22 | 21 | ex 403 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝐿 ∈ (0...(𝑁 − 𝑀)) → ((𝑊 substr 〈𝑀, 𝑁〉) prefix 𝐿) = (𝑊 substr 〈𝑀, (𝑀 + 𝐿)〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 〈cop 4404 ‘cfv 6137 (class class class)co 6924 0cc0 10274 + caddc 10277 − cmin 10608 ℕ0cn0 11647 ...cfz 12648 ♯chash 13441 Word cword 13605 substr csubstr 13736 prefix cpfx 13785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-substr 13737 df-pfx 13786 |
This theorem is referenced by: pfxpfx 13826 |
Copyright terms: Public domain | W3C validator |