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Theorem pfxval 14561
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 14559 . . 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 4831 . . . . 5 (𝑙 = 𝐿 → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
54adantl 482 . . . 4 ((𝑠 = 𝑆𝑙 = 𝐿) → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
63, 5oveq12d 7375 . . 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 3463 . . 3 (𝑆𝑉𝑆 ∈ V)
98adantr 481 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝑆 ∈ V)
10 simpr 485 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0)
11 ovexd 7392 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 substr ⟨0, 𝐿⟩) ∈ V)
122, 7, 9, 10, 11ovmpod 7507 1 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cop 4592  (class class class)co 7357  cmpo 7359  0cc0 11051  0cn0 12413   substr csubstr 14528   prefix cpfx 14558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-pfx 14559
This theorem is referenced by:  pfx00  14562  pfx0  14563  pfxval0  14564  pfxcl  14565  pfxmpt  14566  pfxfv  14570  pfxnd  14575  pfx1  14591  pfxswrd  14594  swrdpfx  14595  pfxpfx  14596  swrdccat  14623  pfxccatpfx1  14624  pfxccatpfx2  14625  cshw0  14682  pfxco  14727  clwwlkf1  28993  cycpmco2f1  31973
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