Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pfxval Structured version   Visualization version   GIF version

Theorem pfxval 40246
 Description: Value of a prefix. (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 40245 . . 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 472 . . . 4 ((𝑠 = 𝑆𝑙 = 𝐿) → 𝑠 = 𝑆)
4 opeq2 4341 . . . . 5 (𝑙 = 𝐿 → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
54adantl 481 . . . 4 ((𝑠 = 𝑆𝑙 = 𝐿) → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
63, 5oveq12d 6567 . . 3 ((𝑠 = 𝑆𝑙 = 𝐿) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
76adantl 481 . 2 (((𝑆𝑉𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆𝑙 = 𝐿)) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
8 elex 3185 . . 3 (𝑆𝑉𝑆 ∈ V)
98adantr 480 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝑆 ∈ V)
10 simpr 476 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0)
11 ovex 6577 . . 3 (𝑆 substr ⟨0, 𝐿⟩) ∈ V
1211a1i 11 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 substr ⟨0, 𝐿⟩) ∈ V)
132, 7, 9, 10, 12ovmpt2d 6686 1 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  ℕ0cn0 11169   substr csubstr 13150   prefix cpfx 40244 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pfx 40245 This theorem is referenced by:  pfx00  40247  pfx0  40248  pfxcl  40249  pfxmpt  40250  pfxid  40255  pfxn0  40257  pfxnd  40258  pfxfv  40262  pfx1  40274  pfxswrd  40276  swrdpfx  40277  pfxpfx  40278  pfxccatpfx1  40290  pfxccatpfx2  40291  splvalpfx  40298  cshword2  40300  pfxco  40301
 Copyright terms: Public domain W3C validator