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Theorem swrdnznd 14650
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 6936). (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.
StepHypRef Expression
1 opelxp 5718 . . . 4 (⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ) ↔ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
21biimpi 215 . . 3 (⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
32adantl 480 . 2 ((𝑆 ∈ V ∧ ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ)) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
4 df-substr 14649 . . 3 substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅))
54mpondm0 7666 . 2 (¬ (𝑆 ∈ V ∧ ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
63, 5nsyl5 159 1 (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  wss 3947  c0 4325  ifcif 4533  cop 4639  cmpt 5236   × cxp 5680  dom cdm 5682  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  0cc0 11158   + caddc 11161  cmin 11494  cz 12610  ..^cfzo 13681   substr csubstr 14648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-dm 5692  df-iota 6506  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-substr 14649
This theorem is referenced by:  swrdnnn0nd  14664
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