swrdccat3a - Metamath Proof Explorer

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Theorem swrdccat3a 13345

Description: A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 29-May-2018.)

**Hypothesis**
Ref	Expression
swrdccatin12.l	⊢ 𝐿 = (#‘𝐴)

**Assertion**
Ref	Expression
swrdccat3a	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))

**Proof of Theorem swrdccat3a**
Step	Hyp	Ref	Expression
1		elfznn0 12302	. . . . . 6 ⊢ (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℕ₀)
2		0elfz 12305	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
3	1, 2	syl 17	. . . . 5 ⊢ (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → 0 ∈ (0...𝑁))
4	3	ancri 573	. . . 4 ⊢ (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → (0 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))))
5		swrdccatin12.l	. . . . . 6 ⊢ 𝐿 = (#‘𝐴)
6	5	swrdccat3 13343	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((0 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), if(𝐿 ≤ 0, (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨0, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))))
7	6	imp 444	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (0 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), if(𝐿 ≤ 0, (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨0, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))
8	4, 7	sylan2 490	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), if(𝐿 ≤ 0, (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨0, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))
9		iftrue 4042	. . . . 5 ⊢ (𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = (𝐴 substr ⟨0, 𝑁⟩))
10	9	adantl 481	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ 𝑁 ≤ 𝐿) → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = (𝐴 substr ⟨0, 𝑁⟩))
11		iffalse 4045	. . . . . 6 ⊢ (¬ 𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))
12	11	3ad2ant2 1076	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 0) → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))
13		lencl 13179	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ₀)
14	5, 13	syl5eqel 2692	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℕ₀)
15		nn0le0eq0 11198	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ₀ → (𝐿 ≤ 0 ↔ 𝐿 = 0))
16	14, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿 ≤ 0 ↔ 𝐿 = 0))
17	16	biimpd 218	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿 ≤ 0 → 𝐿 = 0))
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐿 ≤ 0 → 𝐿 = 0))
19	5	eqeq1i 2615	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 = 0 ↔ (#‘𝐴) = 0)
20	19	biimpi 205	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = 0 → (#‘𝐴) = 0)
21		hasheq0 13015	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
22	20, 21	syl5ib 233	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿 = 0 → 𝐴 = ∅))
23	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐿 = 0 → 𝐴 = ∅))
24	23	imp 444	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → 𝐴 = ∅)
25		0m0e0 11007	. . . . . . . . . . . . . . . 16 ⊢ (0 − 0) = 0
26		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 𝐿 → (0 − 0) = (0 − 𝐿))
27	26	eqcoms 2618	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 = 0 → (0 − 0) = (0 − 𝐿))
28	25, 27	syl5eqr 2658	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = 0 → 0 = (0 − 𝐿))
29	28	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → 0 = (0 − 𝐿))
30	29	opeq1d 4346	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → ⟨0, (𝑁 − 𝐿)⟩ = ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)
31	30	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
32	24, 31	oveq12d 6567	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (∅ ++ (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)))
33		swrdcl 13271	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩) ∈ Word 𝑉)
34		ccatlid 13222	. . . . . . . . . . . . . 14 ⊢ ((𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩) ∈ Word 𝑉 → (∅ ++ (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
35	33, 34	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝑉 → (∅ ++ (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
36	35	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (∅ ++ (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
37	36	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → (∅ ++ (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
38	32, 37	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
39	38	ex 449	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐿 = 0 → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)))
40	18, 39	syld 46	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐿 ≤ 0 → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)))
41	40	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿 ≤ 0 → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)))
42	41	imp 444	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ 𝐿 ≤ 0) → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
43	42	3adant2 1073	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 0) → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
44	12, 43	eqtrd 2644	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 0) → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
45	11	3ad2ant2 1076	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 0) → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))
46	5	opeq2i 4344	. . . . . . . . . . 11 ⊢ ⟨0, 𝐿⟩ = ⟨0, (#‘𝐴)⟩
47	46	oveq2i 6560	. . . . . . . . . 10 ⊢ (𝐴 substr ⟨0, 𝐿⟩) = (𝐴 substr ⟨0, (#‘𝐴)⟩)
48		swrdid 13280	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
49	47, 48	syl5req 2657	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
50	49	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
51	50	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
52	51	3ad2ant1 1075	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 0) → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
53	52	oveq1d 6564	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 0) → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = ((𝐴 substr ⟨0, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))
54	45, 53	eqtrd 2644	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 0) → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = ((𝐴 substr ⟨0, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))
55	10, 44, 54	2if2 4086	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), if(𝐿 ≤ 0, (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨0, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))
56	8, 55	eqtr4d 2647	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))
57	56	ex 449	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ifcif 4036 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕ₀cn0 11169 ...cfz 12197 #chash 12979 Word cword 13146 ++ cconcat 13148 substr csubstr 13150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158

This theorem is referenced by: swrdccatid 13348

W3C validator