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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 6942). (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 5725 | . . . 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 14676 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ifcif 4531 〈cop 4637 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 0cc0 11153 + caddc 11156 − cmin 11490 ℤcz 12611 ..^cfzo 13691 substr csubstr 14675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-substr 14676 |
This theorem is referenced by: swrdnnn0nd 14691 |
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