Theorem swrdnznd 14605

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 6872). (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 5667	. . . 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 14604	. . 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 7607	. 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 1542   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  ifcif 4466  ⟨cop 4573   ↦ cmpt 5166   × cxp 5629  dom cdm 5631  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  0cc0 11038   + caddc 11041   − cmin 11377  ℤcz 12524  ..^cfzo 13608   substr csubstr 14603

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-dm 5641  df-iota 6454  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-substr 14604

This theorem is referenced by:  swrdnnn0nd  14619