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Mirrors > Home > MPE Home > Th. List > Mathboxes > pfxval | Structured version Visualization version GIF version |
Description: Value of a prefix. (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 41382 | . . 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 473 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 𝑠 = 𝑆) | |
4 | opeq2 4403 | . . . . 5 ⊢ (𝑙 = 𝐿 → 〈0, 𝑙〉 = 〈0, 𝐿〉) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 〈0, 𝑙〉 = 〈0, 𝐿〉) |
6 | 3, 5 | oveq12d 6668 | . . 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 3212 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
9 | 8 | adantr 481 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝑆 ∈ V) |
10 | simpr 477 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0) | |
11 | ovexd 6680 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 substr 〈0, 𝐿〉) ∈ V) | |
12 | 2, 7, 9, 10, 11 | ovmpt2d 6788 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr 〈0, 𝐿〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 (class class class)co 6650 ↦ cmpt2 6652 0cc0 9936 ℕ0cn0 11292 substr csubstr 13295 prefix cpfx 41381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pfx 41382 |
This theorem is referenced by: pfx00 41384 pfx0 41385 pfxcl 41386 pfxmpt 41387 pfxid 41392 pfxn0 41394 pfxnd 41395 pfxfv 41399 pfx1 41411 pfxswrd 41413 swrdpfx 41414 pfxpfx 41415 pfxccatpfx1 41427 pfxccatpfx2 41428 splvalpfx 41435 cshword2 41437 pfxco 41438 |
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