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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 6725). (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 5572 | . . . 4 ⊢ (〈𝐹, 𝐿〉 ∈ (ℤ × ℤ) ↔ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) | |
2 | 1 | biimpi 219 | . . 3 ⊢ (〈𝐹, 𝐿〉 ∈ (ℤ × ℤ) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) |
3 | 2 | adantl 485 | . 2 ⊢ ((𝑆 ∈ V ∧ 〈𝐹, 𝐿〉 ∈ (ℤ × ℤ)) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) |
4 | df-substr 14171 | . . 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 7424 | . 2 ⊢ (¬ (𝑆 ∈ V ∧ 〈𝐹, 𝐿〉 ∈ (ℤ × ℤ)) → (𝑆 substr 〈𝐹, 𝐿〉) = ∅) |
6 | 3, 5 | nsyl5 162 | 1 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr 〈𝐹, 𝐿〉) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 ifcif 4425 〈cop 4533 ↦ cmpt 5120 × cxp 5534 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 0cc0 10694 + caddc 10697 − cmin 11027 ℤcz 12141 ..^cfzo 13203 substr csubstr 14170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-dm 5546 df-iota 6316 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-substr 14171 |
This theorem is referenced by: swrdnnn0nd 14186 |
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