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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 14627 | . . 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 481 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 𝑠 = 𝑆) | |
4 | opeq2 4875 | . . . . 5 ⊢ (𝑙 = 𝐿 → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩) | |
5 | 4 | adantl 480 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩) |
6 | 3, 5 | oveq12d 7431 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩)) |
7 | 6 | adantl 480 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆 ∧ 𝑙 = 𝐿)) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩)) |
8 | elex 3491 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
9 | 8 | adantr 479 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝑆 ∈ V) |
10 | simpr 483 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0) | |
11 | ovexd 7448 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 substr ⟨0, 𝐿⟩) ∈ V) | |
12 | 2, 7, 9, 10, 11 | ovmpod 7564 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ⟨cop 4635 (class class class)co 7413 ∈ cmpo 7415 0cc0 11114 ℕ0cn0 12478 substr csubstr 14596 prefix cpfx 14626 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-pfx 14627 |
This theorem is referenced by: pfx00 14630 pfx0 14631 pfxval0 14632 pfxcl 14633 pfxmpt 14634 pfxfv 14638 pfxnd 14643 pfx1 14659 pfxswrd 14662 swrdpfx 14663 pfxpfx 14664 swrdccat 14691 pfxccatpfx1 14692 pfxccatpfx2 14693 cshw0 14750 pfxco 14795 clwwlkf1 29567 cycpmco2f1 32551 |
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