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Mirrors > Home > MPE Home > Th. List > swrdnznd | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
swrdnznd | ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr 〈𝐹, 𝐿〉) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5667 | . . . 4 ⊢ (〈𝐹, 𝐿〉 ∈ (ℤ × ℤ) ↔ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (〈𝐹, 𝐿〉 ∈ (ℤ × ℤ) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) |
3 | 2 | adantl 482 | . 2 ⊢ ((𝑆 ∈ V ∧ 〈𝐹, 𝐿〉 ∈ (ℤ × ℤ)) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) |
4 | df-substr 14521 | . . 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 7590 | . 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〉) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 ∅c0 4280 ifcif 4484 〈cop 4590 ↦ cmpt 5186 × cxp 5629 dom cdm 5631 ‘cfv 6493 (class class class)co 7353 1st c1st 7915 2nd c2nd 7916 0cc0 11047 + caddc 11050 − cmin 11381 ℤcz 12495 ..^cfzo 13559 substr csubstr 14520 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-dm 5641 df-iota 6445 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-substr 14521 |
This theorem is referenced by: swrdnnn0nd 14536 |
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