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Mirrors > Home > MPE Home > Th. List > pfxval | Structured version Visualization version GIF version |
Description: Value of a prefix operation. (Contributed by AV, 2-May-2020.) |
Ref | Expression |
---|---|
pfxval | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr 〈0, 𝐿〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pfx 14021 | . . 3 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉)) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉))) |
3 | simpl 483 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 𝑠 = 𝑆) | |
4 | opeq2 4796 | . . . . 5 ⊢ (𝑙 = 𝐿 → 〈0, 𝑙〉 = 〈0, 𝐿〉) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 〈0, 𝑙〉 = 〈0, 𝐿〉) |
6 | 3, 5 | oveq12d 7163 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → (𝑠 substr 〈0, 𝑙〉) = (𝑆 substr 〈0, 𝐿〉)) |
7 | 6 | adantl 482 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆 ∧ 𝑙 = 𝐿)) → (𝑠 substr 〈0, 𝑙〉) = (𝑆 substr 〈0, 𝐿〉)) |
8 | elex 3510 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
9 | 8 | adantr 481 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝑆 ∈ V) |
10 | simpr 485 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0) | |
11 | ovexd 7180 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 substr 〈0, 𝐿〉) ∈ V) | |
12 | 2, 7, 9, 10, 11 | ovmpod 7291 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr 〈0, 𝐿〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 (class class class)co 7145 ∈ cmpo 7147 0cc0 10525 ℕ0cn0 11885 substr csubstr 13990 prefix cpfx 14020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-pfx 14021 |
This theorem is referenced by: pfx00 14024 pfx0 14025 pfxval0 14026 pfxcl 14027 pfxmpt 14028 pfxfv 14032 pfxnd 14037 pfx1 14053 pfxswrd 14056 swrdpfx 14057 pfxpfx 14058 swrdccat 14085 pfxccatpfx1 14086 pfxccatpfx2 14087 cshw0 14144 pfxco 14188 clwwlkf1 27755 cycpmco2f1 30693 |
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