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Theorem pfxval 14023
Description: Value of a prefix operation. (Contributed by AV, 2-May-2020.)
Assertion
Ref Expression
pfxval ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))

Proof of Theorem pfxval
Dummy variables 𝑙 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pfx 14021 . . 3 prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
21a1i 11 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩)))
3 simpl 483 . . . 4 ((𝑠 = 𝑆𝑙 = 𝐿) → 𝑠 = 𝑆)
4 opeq2 4796 . . . . 5 (𝑙 = 𝐿 → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
54adantl 482 . . . 4 ((𝑠 = 𝑆𝑙 = 𝐿) → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
63, 5oveq12d 7163 . . 3 ((𝑠 = 𝑆𝑙 = 𝐿) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
76adantl 482 . 2 (((𝑆𝑉𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆𝑙 = 𝐿)) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
8 elex 3510 . . 3 (𝑆𝑉𝑆 ∈ V)
98adantr 481 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝑆 ∈ V)
10 simpr 485 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0)
11 ovexd 7180 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 substr ⟨0, 𝐿⟩) ∈ V)
122, 7, 9, 10, 11ovmpod 7291 1 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cop 4563  (class class class)co 7145  cmpo 7147  0cc0 10525  0cn0 11885   substr csubstr 13990   prefix cpfx 14020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pfx 14021
This theorem is referenced by:  pfx00  14024  pfx0  14025  pfxval0  14026  pfxcl  14027  pfxmpt  14028  pfxfv  14032  pfxnd  14037  pfx1  14053  pfxswrd  14056  swrdpfx  14057  pfxpfx  14058  swrdccat  14085  pfxccatpfx1  14086  pfxccatpfx2  14087  cshw0  14144  pfxco  14188  clwwlkf1  27755  cycpmco2f1  30693
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