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Mirrors > Home > MPE Home > Th. List > pfxfv | Structured version Visualization version GIF version |
Description: A symbol in a prefix of a word, indexed using the prefix' indices. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.) |
Ref | Expression |
---|---|
pfxfv | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = (𝑊‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznn0 13001 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℕ0) | |
2 | pfxval 14035 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑊 prefix 𝐿) = (𝑊 substr 〈0, 𝐿〉)) | |
3 | 1, 2 | sylan2 594 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) = (𝑊 substr 〈0, 𝐿〉)) |
4 | 3 | 3adant3 1128 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → (𝑊 prefix 𝐿) = (𝑊 substr 〈0, 𝐿〉)) |
5 | 4 | fveq1d 6672 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = ((𝑊 substr 〈0, 𝐿〉)‘𝐼)) |
6 | simp1 1132 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 𝑊 ∈ Word 𝑉) | |
7 | 0elfz 13005 | . . . . 5 ⊢ (𝐿 ∈ ℕ0 → 0 ∈ (0...𝐿)) | |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 0 ∈ (0...𝐿)) |
9 | 8 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 0 ∈ (0...𝐿)) |
10 | simp2 1133 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 𝐿 ∈ (0...(♯‘𝑊))) | |
11 | 1 | nn0cnd 11958 | . . . . . . . . . 10 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℂ) |
12 | 11 | subid1d 10986 | . . . . . . . . 9 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (𝐿 − 0) = 𝐿) |
13 | 12 | eqcomd 2827 | . . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 = (𝐿 − 0)) |
14 | 13 | oveq2d 7172 | . . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (0..^𝐿) = (0..^(𝐿 − 0))) |
15 | 14 | eleq2d 2898 | . . . . . 6 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (𝐼 ∈ (0..^𝐿) ↔ 𝐼 ∈ (0..^(𝐿 − 0)))) |
16 | 15 | biimpd 231 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ (0..^(𝐿 − 0)))) |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (𝐿 ∈ (0...(♯‘𝑊)) → (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ (0..^(𝐿 − 0))))) |
18 | 17 | 3imp 1107 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 𝐼 ∈ (0..^(𝐿 − 0))) |
19 | swrdfv 14010 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) ∧ 𝐼 ∈ (0..^(𝐿 − 0))) → ((𝑊 substr 〈0, 𝐿〉)‘𝐼) = (𝑊‘(𝐼 + 0))) | |
20 | 6, 9, 10, 18, 19 | syl31anc 1369 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 substr 〈0, 𝐿〉)‘𝐼) = (𝑊‘(𝐼 + 0))) |
21 | elfzoelz 13039 | . . . . . 6 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ) | |
22 | 21 | zcnd 12089 | . . . . 5 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℂ) |
23 | 22 | addid1d 10840 | . . . 4 ⊢ (𝐼 ∈ (0..^𝐿) → (𝐼 + 0) = 𝐼) |
24 | 23 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → (𝐼 + 0) = 𝐼) |
25 | 24 | fveq2d 6674 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → (𝑊‘(𝐼 + 0)) = (𝑊‘𝐼)) |
26 | 5, 20, 25 | 3eqtrd 2860 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = (𝑊‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4573 ‘cfv 6355 (class class class)co 7156 0cc0 10537 + caddc 10540 − cmin 10870 ℕ0cn0 11898 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Word cword 13862 substr csubstr 14002 prefix cpfx 14032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-substr 14003 df-pfx 14033 |
This theorem is referenced by: pfxid 14046 pfxfv0 14054 pfxtrcfv 14055 pfxfvlsw 14057 pfxeq 14058 ccatpfx 14063 pfxccatin12lem2 14093 splfv1 14117 repswpfx 14147 cshwidxmod 14165 pfx2 14309 wwlksm1edg 27659 wwlksnred 27670 clwwlkinwwlk 27818 clwwlkf 27826 wwlksubclwwlk 27837 dlwwlknondlwlknonf1olem1 28143 cycpmco2 30775 revpfxsfxrev 32362 |
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