Theorem swrdnznd 14676

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 5695	. . 3 ⊢ (⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ) ↔ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
2	1	bilani 509	. 2 ⊢ ((𝑆 ∈ V ∧ ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ)) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
3		df-substr 14675	. . 3 ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))
4	3	mpondm0 7648	. 2 ⊢ (¬ (𝑆 ∈ V ∧ ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
5	2, 4	nsyl5 160	1 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ifcif 4489  ⟨cop 4597   ↦ cmpt 5193   × cxp 5657  dom cdm 5659  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  0cc0 11096   + caddc 11099   − cmin 11437  ℤcz 12587  ..^cfzo 13678   substr csubstr 14674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5665  df-dm 5669  df-iota 6490  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-substr 14675

This theorem is referenced by:  swrdnnn0nd  14690