Theorem swrdnnn0nd 14003

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 978	. . . 4 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
2		ianor 978	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℤ ∧ 0 ≤ 𝐹) ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹))
3		elnn0z 11981	. . . . . . 7 ⊢ (𝐹 ∈ ℕ0 ↔ (𝐹 ∈ ℤ ∧ 0 ≤ 𝐹))
4	2, 3	xchnxbir 335	. . . . . 6 ⊢ (¬ 𝐹 ∈ ℕ0 ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹))
5		ianor 978	. . . . . . 7 ⊢ (¬ (𝐿 ∈ ℤ ∧ 0 ≤ 𝐿) ↔ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿))
6		elnn0z 11981	. . . . . . 7 ⊢ (𝐿 ∈ ℕ0 ↔ (𝐿 ∈ ℤ ∧ 0 ≤ 𝐿))
7	5, 6	xchnxbir 335	. . . . . 6 ⊢ (¬ 𝐿 ∈ ℕ0 ↔ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿))
8	4, 7	orbi12i 911	. . . . 5 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ↔ ((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)))
9		or4 923	. . . . . 6 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)) ↔ ((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
10		ianor 978	. . . . . . . 8 ⊢ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ (¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ))
11	10	bicomi 226	. . . . . . 7 ⊢ ((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ↔ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
12	11	orbi1i 910	. . . . . 6 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
13	9, 12	bitri 277	. . . . 5 ⊢ (((¬ 𝐹 ∈ ℤ ∨ ¬ 0 ≤ 𝐹) ∨ (¬ 𝐿 ∈ ℤ ∨ ¬ 0 ≤ 𝐿)) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
14	8, 13	bitri 277	. . . 4 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
15	1, 14	bitri 277	. . 3 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)))
16		swrdnznd 13989	. . . . 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 317	. . . . . 6 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ ¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
19		zre 11972	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ ℤ → 𝐹 ∈ ℝ)
20		0red 10630	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ ℤ → 0 ∈ ℝ)
21	19, 20	jca 514	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ ℤ → (𝐹 ∈ ℝ ∧ 0 ∈ ℝ))
22	21	3ad2ant2 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 ∈ ℝ ∧ 0 ∈ ℝ))
23		ltnle 10706	. . . . . . . . . . . . . . . . . 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 863	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 < 0 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
26	24, 25	syl6bir 256	. . . . . . . . . . . . . . . 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 317	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ≤ 𝐹 ↔ ¬ ¬ 0 ≤ 𝐹)
29	28	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (0 ≤ 𝐹 ↔ ¬ ¬ 0 ≤ 𝐹))
30		zre 11972	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
31		0red 10630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℤ → 0 ∈ ℝ)
32	30, 31	jca 514	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℤ → (𝐿 ∈ ℝ ∧ 0 ∈ ℝ))
33	32	3ad2ant3 1131	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 0 ∈ ℝ))
34		ltnle 10706	. . . . . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((0 ≤ 𝐹 ∧ 𝐿 < 0) ↔ (¬ ¬ 0 ≤ 𝐹 ∧ ¬ 0 ≤ 𝐿)))
37	30	3ad2ant3 1131	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐿 ∈ ℝ)
38		0red 10630	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 0 ∈ ℝ)
39	19	3ad2ant2 1130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐹 ∈ ℝ)
40		ltleletr 10719	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐹 ∈ ℝ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → 𝐿 ≤ 𝐹))
41	37, 38, 39, 40	syl3anc 1367	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐿 < 0 ∧ 0 ≤ 𝐹) → 𝐿 ≤ 𝐹))
42		olc 864	. . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((0 ≤ 𝐹 ∧ 𝐿 < 0) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
45	36, 44	sylbird 262	. . . . . . . . . . . . . . . 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 1055	. . . . . . . . . . . . . 14 ⊢ ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹)))
48	47	impcom 410	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹))
49	48	orcd 869	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹) ∨ (♯‘𝑆) < 𝐿))
50		df-3or 1084	. . . . . . . . . . . 12 ⊢ ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) ↔ ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹) ∨ (♯‘𝑆) < 𝐿))
51	49, 50	sylibr 236	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
52		swrdnd 14001	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
53	52	imp 409	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
54	51, 53	syldan 593	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
55	54	ex 415	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
56	55	3expb 1116	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
57	56	expcom 416	. . . . . . 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 237	. . . . 5 ⊢ (¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
60	59	imp 409	. . . 4 ⊢ ((¬ ¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
61	17, 60	jaoi3 1055	. . 3 ⊢ ((¬ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∨ (¬ 0 ≤ 𝐹 ∨ ¬ 0 ≤ 𝐿)) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
62	15, 61	sylbi 219	. 2 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
63	62	impcom 410	1 ⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∅c0 4279  ⟨cop 4559   class class class wbr 5052  ‘cfv 6341  (class class class)co 7142  ℝcr 10522  0cc0 10523   < clt 10661   ≤ cle 10662  ℕ₀cn0 11884  ℤcz 11968  ♯chash 13680  Word cword 13851   substr csubstr 13987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447  ax-cnex 10579  ax-resscn 10580  ax-1cn 10581  ax-icn 10582  ax-addcl 10583  ax-addrcl 10584  ax-mulcl 10585  ax-mulrcl 10586  ax-mulcom 10587  ax-addass 10588  ax-mulass 10589  ax-distr 10590  ax-i2m1 10591  ax-1ne0 10592  ax-1rid 10593  ax-rnegex 10594  ax-rrecex 10595  ax-cnre 10596  ax-pre-lttri 10597  ax-pre-lttrn 10598  ax-pre-ltadd 10599  ax-pre-mulgt0 10600

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-pss 3942  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-tp 4558  df-op 4560  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5446  df-eprel 5451  df-po 5460  df-so 5461  df-fr 5500  df-we 5502  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-pred 6134  df-ord 6180  df-on 6181  df-lim 6182  df-suc 6183  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7567  df-1st 7675  df-2nd 7676  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-card 9354  df-pnf 10663  df-mnf 10664  df-xr 10665  df-ltxr 10666  df-le 10667  df-sub 10858  df-neg 10859  df-nn 11625  df-n0 11885  df-z 11969  df-uz 12231  df-fz 12883  df-fzo 13024  df-hash 13681  df-word 13852  df-substr 13988

This theorem is referenced by:  swrdnd0  14004  pfxval0  14023