Theorem swrdnnn0nd 13721

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 1011	. . . 4 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
2		ianor 1011	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℤ ∧ 0 ≤ 𝐹) ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹))
3		elnn0z 11717	. . . . . . 7 ⊢ (𝐹 ∈ ℕ0 ↔ (𝐹 ∈ ℤ ∧ 0 ≤ 𝐹))
4	2, 3	xchnxbir 325	. . . . . 6 ⊢ (¬ 𝐹 ∈ ℕ0 ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹))
5		ianor 1011	. . . . . . 7 ⊢ (¬ (𝐿 ∈ ℤ ∧ 0 ≤ 𝐿) ↔ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿))
6		elnn0z 11717	. . . . . . 7 ⊢ (𝐿 ∈ ℕ0 ↔ (𝐿 ∈ ℤ ∧ 0 ≤ 𝐿))
7	5, 6	xchnxbir 325	. . . . . 6 ⊢ (¬ 𝐿 ∈ ℕ0 ↔ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿))
8	4, 7	orbi12i 945	. . . . 5 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ↔ ((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)))
9		or4 957	. . . . . 6 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)) ↔ ((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
10		ianor 1011	. . . . . . . 8 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ))
11	10	bicomi 216	. . . . . . 7 ⊢ ((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ↔ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
12	11	orbi1i 944	. . . . . 6 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
13	9, 12	bitri 267	. . . . 5 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
14	8, 13	bitri 267	. . . 4 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
15	1, 14	bitri 267	. . 3 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
16		swrdnznd 13702	. . . . 5 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
17	16	a1d 25	. . . 4 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
18		notnotb 307	. . . . . 6 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ ¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
19		zre 11708	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ ℤ → 𝐹 ∈ ℝ)
20		0red 10360	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ ℤ → 0 ∈ ℝ)
21	19, 20	jca 509	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ ℤ → (𝐹 ∈ ℝ ∧ 0 ∈ ℝ))
22	21	3ad2ant2 1170	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 ∈ ℝ ∧ 0 ∈ ℝ))
23		ltnle 10436	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐹 < 0 ↔ ¬ 0 ≤ 𝐹))
24	22, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ↔ ¬ 0 ≤ 𝐹))
25		orc 900	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 < 0 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
26	24, 25	syl6bir 246	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (¬ 0 ≤ 𝐹 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
27	26	com12 32	. . . . . . . . . . . . . . 15 ⊢ (¬ 0 ≤ 𝐹 → ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
28		notnotb 307	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ≤ 𝐹 ↔ ¬ ¬ 0 ≤ 𝐹)
29	28	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (0 ≤ 𝐹 ↔ ¬ ¬ 0 ≤ 𝐹))
30		zre 11708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
31		0red 10360	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℤ → 0 ∈ ℝ)
32	30, 31	jca 509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℤ → (𝐿 ∈ ℝ ∧ 0 ∈ ℝ))
33	32	3ad2ant3 1171	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 0 ∈ ℝ))
34		ltnle 10436	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐿 < 0 ↔ ¬ 0 ≤ 𝐿))
35	33, 34	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 < 0 ↔ ¬ 0 ≤ 𝐿))
36	29, 35	anbi12d 626	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((0 ≤ 𝐹 ∧ 𝐿 < 0) ↔ (¬ ¬ 0 ≤ 𝐹 ∧ ¬ 0 ≤ 𝐿)))
37	30	3ad2ant3 1171	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐿 ∈ ℝ)
38		0red 10360	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 0 ∈ ℝ)
39	19	3ad2ant2 1170	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐹 ∈ ℝ)
40		ltleletr 10449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐹 ∈ ℝ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → 𝐿 ≤ 𝐹))
41	37, 38, 39, 40	syl3anc 1496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → 𝐿 ≤ 𝐹))
42		olc 901	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ≤ 𝐹 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
43	41, 42	syl6 35	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
44	43	ancomsd 459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((0 ≤ 𝐹 ∧ 𝐿 < 0) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
45	36, 44	sylbird 252	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ ¬ 0 ≤ 𝐹 ∧ ¬ 0 ≤ 𝐿) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
46	45	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((¬ ¬ 0 ≤ 𝐹 ∧ ¬ 0 ≤ 𝐿) → ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
47	27, 46	jaoi3 1089	. . . . . . . . . . . . . 14 ⊢ ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
48	47	impcom 398	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
49	48	orcd 906	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹) ∨ (♯‘𝑆) < 𝐿))
50		df-3or 1114	. . . . . . . . . . . 12 ⊢ ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) ↔ ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹) ∨ (♯‘𝑆) < 𝐿))
51	49, 50	sylibr 226	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
52		swrdnd 13719	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
53	52	imp 397	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
54	51, 53	syldan 587	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
55	54	ex 403	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
56	55	3expb 1155	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
57	56	expcom 404	. . . . . . 7 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 ∈ Word 𝑉 → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
58	57	com23 86	. . . . . 6 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
59	18, 58	sylbir 227	. . . . 5 ⊢ (¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
60	59	imp 397	. . . 4 ⊢ ((¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
61	17, 60	jaoi3 1089	. . 3 ⊢ ((¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
62	15, 61	sylbi 209	. 2 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
63	62	impcom 398	1 ⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 880   ∨ w3o 1112   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∅c0 4144  ⟨cop 4403   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  ℝcr 10251  0cc0 10252   < clt 10391   ≤ cle 10392  ℕ₀cn0 11618  ℤcz 11704  ♯chash 13410  Word cword 13574   substr csubstr 13700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-hash 13411  df-word 13575  df-substr 13701

This theorem is referenced by:  swrdnd0  13722  pfxval0  13755