![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13576 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℕ0) | |
2 | pfxval 14605 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑊 prefix 𝐿) = (𝑊 substr 〈0, 𝐿〉)) | |
3 | 1, 2 | sylan2 593 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) = (𝑊 substr 〈0, 𝐿〉)) |
4 | 3 | 3adant3 1132 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → (𝑊 prefix 𝐿) = (𝑊 substr 〈0, 𝐿〉)) |
5 | 4 | fveq1d 6880 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = ((𝑊 substr 〈0, 𝐿〉)‘𝐼)) |
6 | simp1 1136 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 𝑊 ∈ Word 𝑉) | |
7 | 0elfz 13580 | . . . . 5 ⊢ (𝐿 ∈ ℕ0 → 0 ∈ (0...𝐿)) | |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 0 ∈ (0...𝐿)) |
9 | 8 | 3ad2ant2 1134 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 0 ∈ (0...𝐿)) |
10 | simp2 1137 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 𝐿 ∈ (0...(♯‘𝑊))) | |
11 | 1 | nn0cnd 12516 | . . . . . . . . . 10 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℂ) |
12 | 11 | subid1d 11542 | . . . . . . . . 9 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (𝐿 − 0) = 𝐿) |
13 | 12 | eqcomd 2737 | . . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 = (𝐿 − 0)) |
14 | 13 | oveq2d 7409 | . . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (0..^𝐿) = (0..^(𝐿 − 0))) |
15 | 14 | eleq2d 2818 | . . . . . 6 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (𝐼 ∈ (0..^𝐿) ↔ 𝐼 ∈ (0..^(𝐿 − 0)))) |
16 | 15 | biimpd 228 | . . . . 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 1111 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → 𝐼 ∈ (0..^(𝐿 − 0))) |
19 | swrdfv 14580 | . . 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 1373 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 substr 〈0, 𝐿〉)‘𝐼) = (𝑊‘(𝐼 + 0))) |
21 | elfzoelz 13614 | . . . . . 6 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ) | |
22 | 21 | zcnd 12649 | . . . . 5 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℂ) |
23 | 22 | addridd 11396 | . . . 4 ⊢ (𝐼 ∈ (0..^𝐿) → (𝐼 + 0) = 𝐼) |
24 | 23 | 3ad2ant3 1135 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → (𝐼 + 0) = 𝐼) |
25 | 24 | fveq2d 6882 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → (𝑊‘(𝐼 + 0)) = (𝑊‘𝐼)) |
26 | 5, 20, 25 | 3eqtrd 2775 | 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 1087 = wceq 1541 ∈ wcel 2106 〈cop 4628 ‘cfv 6532 (class class class)co 7393 0cc0 11092 + caddc 11095 − cmin 11426 ℕ0cn0 12454 ...cfz 13466 ..^cfzo 13609 ♯chash 14272 Word cword 14446 substr csubstr 14572 prefix cpfx 14602 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-fzo 13610 df-hash 14273 df-word 14447 df-substr 14573 df-pfx 14603 |
This theorem is referenced by: pfxid 14616 pfxfv0 14624 pfxtrcfv 14625 pfxfvlsw 14627 pfxeq 14628 ccatpfx 14633 pfxccatin12lem2 14663 splfv1 14687 repswpfx 14717 cshwidxmod 14735 pfx2 14880 wwlksm1edg 29000 wwlksnred 29011 clwwlkinwwlk 29158 clwwlkf 29165 wwlksubclwwlk 29176 dlwwlknondlwlknonf1olem1 29482 cycpmco2 32163 revpfxsfxrev 33937 |
Copyright terms: Public domain | W3C validator |