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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 14236 | . . 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 486 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 𝑠 = 𝑆) | |
4 | opeq2 4785 | . . . . 5 ⊢ (𝑙 = 𝐿 → 〈0, 𝑙〉 = 〈0, 𝐿〉) | |
5 | 4 | adantl 485 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 〈0, 𝑙〉 = 〈0, 𝐿〉) |
6 | 3, 5 | oveq12d 7231 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → (𝑠 substr 〈0, 𝑙〉) = (𝑆 substr 〈0, 𝐿〉)) |
7 | 6 | adantl 485 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆 ∧ 𝑙 = 𝐿)) → (𝑠 substr 〈0, 𝑙〉) = (𝑆 substr 〈0, 𝐿〉)) |
8 | elex 3426 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
9 | 8 | adantr 484 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝑆 ∈ V) |
10 | simpr 488 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0) | |
11 | ovexd 7248 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 substr 〈0, 𝐿〉) ∈ V) | |
12 | 2, 7, 9, 10, 11 | ovmpod 7361 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr 〈0, 𝐿〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 〈cop 4547 (class class class)co 7213 ∈ cmpo 7215 0cc0 10729 ℕ0cn0 12090 substr csubstr 14205 prefix cpfx 14235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-pfx 14236 |
This theorem is referenced by: pfx00 14239 pfx0 14240 pfxval0 14241 pfxcl 14242 pfxmpt 14243 pfxfv 14247 pfxnd 14252 pfx1 14268 pfxswrd 14271 swrdpfx 14272 pfxpfx 14273 swrdccat 14300 pfxccatpfx1 14301 pfxccatpfx2 14302 cshw0 14359 pfxco 14403 clwwlkf1 28132 cycpmco2f1 31110 |
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