![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6936). (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 5718 | . . . 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 480 | . 2 ⊢ ((𝑆 ∈ V ∧ 〈𝐹, 𝐿〉 ∈ (ℤ × ℤ)) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) |
4 | df-substr 14649 | . . 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 7666 | . 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 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 ℤcz 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 |
Copyright terms: Public domain | W3C validator |