Theorem swrdnznd 14660

Description: The value of a subword operation for noninteger arguments is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6911). (Contributed by AV, 2-Dec-2022.) (New usage is discouraged.)

Assertion

Ref	Expression
swrdnznd	⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)

Proof of Theorem swrdnznd

Dummy variables 𝑠 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxp 5690	. . . 4 ⊢ (⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ) ↔ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
2	1	biimpi 216	. . 3 ⊢ (⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
3	2	adantl 481	. 2 ⊢ ((𝑆 ∈ V ∧ ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ)) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
4		df-substr 14659	. . 3 ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))
5	4	mpondm0 7647	. 2 ⊢ (¬ (𝑆 ∈ V ∧ ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
6	3, 5	nsyl5 159	1 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ⊆ wss 3926  ∅c0 4308  ifcif 4500  ⟨cop 4607   ↦ cmpt 5201   × cxp 5652  dom cdm 5654  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  0cc0 11129   + caddc 11132   − cmin 11466  ℤcz 12588  ..^cfzo 13671   substr csubstr 14658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-dm 5664  df-iota 6484  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-substr 14659

This theorem is referenced by:  swrdnnn0nd  14674