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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 7306 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 substr 〈𝑀, 𝑁〉) ∈ V) | |
2 | elfznn0 13348 | . . . 4 ⊢ (𝐿 ∈ (0...(𝑁 − 𝑀)) → 𝐿 ∈ ℕ0) | |
3 | pfxval 14384 | . . . 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 596 | . . 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 13287 | . . . . . . 7 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ0) | |
6 | 5 | 3ad2ant3 1134 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑁 − 𝑀) ∈ ℕ0) |
7 | 0elfz 13352 | . . . . . 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 615 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → (0 ∈ (0...(𝑁 − 𝑀)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀)))) |
10 | swrdswrd 14416 | . . . . 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 407 | . . . 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 591 | . . 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 13348 | . . . . . . . 8 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0) | |
14 | nn0cn 12243 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ) | |
15 | 14 | addid1d 11175 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 0) = 𝑀) |
16 | 13, 15 | syl 17 | . . . . . . 7 ⊢ (𝑀 ∈ (0...𝑁) → (𝑀 + 0) = 𝑀) |
17 | 16 | 3ad2ant3 1134 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑀 + 0) = 𝑀) |
18 | 17 | adantr 481 | . . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → (𝑀 + 0) = 𝑀) |
19 | 18 | opeq1d 4816 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ 𝐿 ∈ (0...(𝑁 − 𝑀))) → 〈(𝑀 + 0), (𝑀 + 𝐿)〉 = 〈𝑀, (𝑀 + 𝐿)〉) |
20 | 19 | oveq2d 7287 | . . 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 2784 | . 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 413 | 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 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 Vcvv 3431 〈cop 4573 ‘cfv 6432 (class class class)co 7271 0cc0 10872 + caddc 10875 − cmin 11205 ℕ0cn0 12233 ...cfz 13238 ♯chash 14042 Word cword 14215 substr csubstr 14351 prefix cpfx 14381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-fzo 13382 df-hash 14043 df-word 14216 df-substr 14352 df-pfx 14382 |
This theorem is referenced by: pfxpfx 14419 |
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