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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 6941). (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 5721 | . . . 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 14679 | . . 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 7673 | . 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ifcif 4525 〈cop 4632 ↦ cmpt 5225 × cxp 5683 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 0cc0 11155 + caddc 11158 − cmin 11492 ℤcz 12613 ..^cfzo 13694 substr csubstr 14678 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-substr 14679 |
| This theorem is referenced by: swrdnnn0nd 14694 |
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