Theorem pfxval 14386

Description: Value of a prefix operation. (Contributed by AV, 2-May-2020.)

Assertion

Ref	Expression
pfxval	⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ 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.

Step	Hyp	Ref	Expression
1		df-pfx 14384	. . 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 483	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → 𝑠 = 𝑆)
4		opeq2 4805	. . . . 5 ⊢ (𝑙 = 𝐿 → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
5	4	adantl 482	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
6	3, 5	oveq12d 7293	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑙 = 𝐿) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
7	6	adantl 482	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆 ∧ 𝑙 = 𝐿)) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
8		elex 3450	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
9	8	adantr 481	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝑆 ∈ V)
10		simpr 485	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0)
11		ovexd 7310	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 substr ⟨0, 𝐿⟩) ∈ V)
12	2, 7, 9, 10, 11	ovmpod 7425	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567  (class class class)co 7275   ∈ cmpo 7277  0cc0 10871  ℕ₀cn0 12233   substr csubstr 14353   prefix cpfx 14383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-pfx 14384

This theorem is referenced by:  pfx00  14387  pfx0  14388  pfxval0  14389  pfxcl  14390  pfxmpt  14391  pfxfv  14395  pfxnd  14400  pfx1  14416  pfxswrd  14419  swrdpfx  14420  pfxpfx  14421  swrdccat  14448  pfxccatpfx1  14449  pfxccatpfx2  14450  cshw0  14507  pfxco  14551  clwwlkf1  28413  cycpmco2f1  31391