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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 14709 | . . 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 4874 | . . . . 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 3501 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 (class class class)co 7431 ∈ cmpo 7433 0cc0 11155 ℕ0cn0 12526 substr csubstr 14678 prefix cpfx 14708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pfx 14709 |
| This theorem is referenced by: pfx00 14712 pfx0 14713 pfxval0 14714 pfxcl 14715 pfxmpt 14716 pfxfv 14720 pfxnd 14725 pfx1 14741 pfxswrd 14744 swrdpfx 14745 pfxpfx 14746 swrdccat 14773 pfxccatpfx1 14774 pfxccatpfx2 14775 cshw0 14832 pfxco 14877 clwwlkf1 30068 cycpmco2f1 33144 |
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