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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 14706 | . . 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 482 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 𝑠 = 𝑆) | |
4 | opeq2 4879 | . . . . 5 ⊢ (𝑙 = 𝐿 → 〈0, 𝑙〉 = 〈0, 𝐿〉) | |
5 | 4 | adantl 481 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 〈0, 𝑙〉 = 〈0, 𝐿〉) |
6 | 3, 5 | oveq12d 7449 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → (𝑠 substr 〈0, 𝑙〉) = (𝑆 substr 〈0, 𝐿〉)) |
7 | 6 | adantl 481 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆 ∧ 𝑙 = 𝐿)) → (𝑠 substr 〈0, 𝑙〉) = (𝑆 substr 〈0, 𝐿〉)) |
8 | elex 3499 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
9 | 8 | adantr 480 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝑆 ∈ V) |
10 | simpr 484 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0) | |
11 | ovexd 7466 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 substr 〈0, 𝐿〉) ∈ V) | |
12 | 2, 7, 9, 10, 11 | ovmpod 7585 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr 〈0, 𝐿〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 (class class class)co 7431 ∈ cmpo 7433 0cc0 11153 ℕ0cn0 12524 substr csubstr 14675 prefix cpfx 14705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pfx 14706 |
This theorem is referenced by: pfx00 14709 pfx0 14710 pfxval0 14711 pfxcl 14712 pfxmpt 14713 pfxfv 14717 pfxnd 14722 pfx1 14738 pfxswrd 14741 swrdpfx 14742 pfxpfx 14743 swrdccat 14770 pfxccatpfx1 14771 pfxccatpfx2 14772 cshw0 14829 pfxco 14874 clwwlkf1 30078 cycpmco2f1 33127 |
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