Theorem swrdnnn0nd 14690

Description: The value of a subword operation for arguments not being nonnegative integers is the empty set. (Contributed by AV, 2-Dec-2022.)

Assertion

Ref	Expression
swrdnnn0nd	⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)

Proof of Theorem swrdnnn0nd

Step	Hyp	Ref	Expression
1		ianor 997	. . . 4 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
2		ianor 997	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℤ ∧ 0 ≤ 𝐹) ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹))
3		elnn0z 12600	. . . . . . 7 ⊢ (𝐹 ∈ ℕ0 ↔ (𝐹 ∈ ℤ ∧ 0 ≤ 𝐹))
4	2, 3	xchnxbir 336	. . . . . 6 ⊢ (¬ 𝐹 ∈ ℕ0 ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹))
5		ianor 997	. . . . . . 7 ⊢ (¬ (𝐿 ∈ ℤ ∧ 0 ≤ 𝐿) ↔ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿))
6		elnn0z 12600	. . . . . . 7 ⊢ (𝐿 ∈ ℕ0 ↔ (𝐿 ∈ ℤ ∧ 0 ≤ 𝐿))
7	5, 6	xchnxbir 336	. . . . . 6 ⊢ (¬ 𝐿 ∈ ℕ0 ↔ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿))
8	4, 7	orbi12i 927	. . . . 5 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ↔ ((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)))
9		or4 939	. . . . . 6 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)) ↔ ((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
10		ianor 997	. . . . . . . 8 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ))
11	10	bicomi 227	. . . . . . 7 ⊢ ((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ↔ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
12	11	orbi1i 926	. . . . . 6 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
13	9, 12	bitri 278	. . . . 5 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
14	8, 13	bitri 278	. . . 4 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
15	1, 14	bitri 278	. . 3 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
16		swrdnznd 14676	. . . . 5 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
17	16	a1d 26	. . . 4 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
18		notnotb 318	. . . . . 6 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ ¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
19		zre 12591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ ℤ → 𝐹 ∈ ℝ)
20		0red 11207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ ℤ → 0 ∈ ℝ)
21	19, 20	jca 520	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ ℤ → (𝐹 ∈ ℝ ∧ 0 ∈ ℝ))
22	21	3ad2ant2 1150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 ∈ ℝ ∧ 0 ∈ ℝ))
23		ltnle 11285	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐹 < 0 ↔ ¬ 0 ≤ 𝐹))
24	22, 23	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ↔ ¬ 0 ≤ 𝐹))
25		orc 880	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 < 0 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
26	24, 25	biimtrrdi 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (¬ 0 ≤ 𝐹 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
27	26	com12 33	. . . . . . . . . . . . . . 15 ⊢ (¬ 0 ≤ 𝐹 → ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
28		notnotb 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ≤ 𝐹 ↔ ¬ ¬ 0 ≤ 𝐹)
29	28	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (0 ≤ 𝐹 ↔ ¬ ¬ 0 ≤ 𝐹))
30		zre 12591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
31		0red 11207	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℤ → 0 ∈ ℝ)
32	30, 31	jca 520	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℤ → (𝐿 ∈ ℝ ∧ 0 ∈ ℝ))
33	32	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 0 ∈ ℝ))
34		ltnle 11285	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐿 < 0 ↔ ¬ 0 ≤ 𝐿))
35	33, 34	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 < 0 ↔ ¬ 0 ≤ 𝐿))
36	29, 35	anbi12d 643	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((0 ≤ 𝐹 ∧ 𝐿 < 0) ↔ (¬ ¬ 0 ≤ 𝐹 ∧ ¬ 0 ≤ 𝐿)))
37	30	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐿 ∈ ℝ)
38		0red 11207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 0 ∈ ℝ)
39	19	3ad2ant2 1150	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐹 ∈ ℝ)
40		ltleletr 11299	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐹 ∈ ℝ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → 𝐿 ≤ 𝐹))
41	37, 38, 39, 40	syl3anc 1396	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → 𝐿 ≤ 𝐹))
42		olc 881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ≤ 𝐹 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
43	41, 42	syl6 36	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
44	43	ancomsd 470	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((0 ≤ 𝐹 ∧ 𝐿 < 0) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
45	36, 44	sylbird 263	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ ¬ 0 ≤ 𝐹 ∧ ¬ 0 ≤ 𝐿) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
46	45	com12 33	. . . . . . . . . . . . . . 15 ⊢ ((¬ ¬ 0 ≤ 𝐹 ∧ ¬ 0 ≤ 𝐿) → ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
47	27, 46	jaoi3 1074	. . . . . . . . . . . . . 14 ⊢ ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
48	47	impcom 412	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
49	48	orcd 886	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹) ∨ (♯‘𝑆) < 𝐿))
50		df-3or 1102	. . . . . . . . . . . 12 ⊢ ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) ↔ ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹) ∨ (♯‘𝑆) < 𝐿))
51	49, 50	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
52		swrdnd 14688	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
53	52	imp 411	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
54	51, 53	syldan 602	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
55	54	ex 417	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
56	55	3expb 1136	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
57	56	expcom 418	. . . . . . 7 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 ∈ Word 𝑉 → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
58	57	com23 87	. . . . . 6 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
59	18, 58	sylbir 238	. . . . 5 ⊢ (¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
60	59	imp 411	. . . 4 ⊢ ((¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
61	17, 60	jaoi3 1074	. . 3 ⊢ ((¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
62	15, 61	sylbi 220	. 2 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
63	62	impcom 412	1 ⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∅c0 4294  ⟨cop 4597   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  ℝcr 11095  0cc0 11096   < clt 11239   ≤ cle 11240  ℕ₀cn0 12500  ℤcz 12587  ♯chash 14362  Word cword 14546   substr csubstr 14674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-n0 12501  df-z 12588  df-uz 12859  df-fz 13532  df-fzo 13679  df-hash 14363  df-word 14547  df-substr 14675

This theorem is referenced by:  swrdnd0  14691  pfxval0  14710