Theorem swrdnznd 14578

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 6874). (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 5668	. . . 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 14577	. . 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 7608	. 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 3442   ⊆ wss 3903  ∅c0 4287  ifcif 4481  ⟨cop 4588   ↦ cmpt 5181   × cxp 5630  dom cdm 5632  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  0cc0 11038   + caddc 11041   − cmin 11376  ℤcz 12500  ..^cfzo 13582   substr csubstr 14576

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-dm 5642  df-iota 6456  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-substr 14577

This theorem is referenced by:  swrdnnn0nd  14592