Theorem swrdccat3a 13494

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 12433	. . . . . 6 ⊢ (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℕ0)
2		0elfz 12436	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
3	1, 2	syl 17	. . . . 5 ⊢ (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → 0 ∈ (0...𝑁))
4	3	ancri 575	. . . 4 ⊢ (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → (0 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))))
5		swrdccatin12.l	. . . . . 6 ⊢ 𝐿 = (#‘𝐴)
6	5	swrdccat3 13492	. . . . 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 445	. . . 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 491	. . 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 4092	. . . . 5 ⊢ (𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = (𝐴 substr ⟨0, 𝑁⟩))
10	9	adantl 482	. . . 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 4095	. . . . . 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 1083	. . . . 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 13324	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
14	5, 13	syl5eqel 2705	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℕ0)
15		nn0le0eq0 11321	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 → (𝐿 ≤ 0 ↔ 𝐿 = 0))
16	14, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿 ≤ 0 ↔ 𝐿 = 0))
17	16	biimpd 219	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿 ≤ 0 → 𝐿 = 0))
18	17	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐿 ≤ 0 → 𝐿 = 0))
19	5	eqeq1i 2627	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 = 0 ↔ (#‘𝐴) = 0)
20	19	biimpi 206	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = 0 → (#‘𝐴) = 0)
21		hasheq0 13154	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
22	20, 21	syl5ib 234	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿 = 0 → 𝐴 = ∅))
23	22	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐿 = 0 → 𝐴 = ∅))
24	23	imp 445	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → 𝐴 = ∅)
25		0m0e0 11130	. . . . . . . . . . . . . . . 16 ⊢ (0 − 0) = 0
26		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 𝐿 → (0 − 0) = (0 − 𝐿))
27	26	eqcoms 2630	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 = 0 → (0 − 0) = (0 − 𝐿))
28	25, 27	syl5eqr 2670	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = 0 → 0 = (0 − 𝐿))
29	28	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → 0 = (0 − 𝐿))
30	29	opeq1d 4408	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → ⟨0, (𝑁 − 𝐿)⟩ = ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)
31	30	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
32	24, 31	oveq12d 6668	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (∅ ++ (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)))
33		swrdcl 13419	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩) ∈ Word 𝑉)
34		ccatlid 13369	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (∅ ++ (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
37	36	adantr 481	. . . . . . . . . . 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 2656	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐿 = 0) → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩))
39	38	ex 450	. . . . . . . . 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 47	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐿 ≤ 0 → (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝐵 substr ⟨(0 − 𝐿), (𝑁 − 𝐿)⟩)))
41	40	adantr 481	. . . . . . 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 445	. . . . . 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 1080	. . . . 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 2656	. . . 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 1083	. . . . 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 4406	. . . . . . . . . . 11 ⊢ ⟨0, 𝐿⟩ = ⟨0, (#‘𝐴)⟩
47	46	oveq2i 6661	. . . . . . . . . 10 ⊢ (𝐴 substr ⟨0, 𝐿⟩) = (𝐴 substr ⟨0, (#‘𝐴)⟩)
48		swrdid 13428	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
49	47, 48	syl5req 2669	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
50	49	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
51	50	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
52	51	3ad2ant1 1082	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 0) → 𝐴 = (𝐴 substr ⟨0, 𝐿⟩))
53	52	oveq1d 6665	. . . . 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 2656	. . . 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 4136	. . 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 2659	. 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 450	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)⟩)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∅c0 3915  ifcif 4086  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  0cc0 9936   + caddc 9939   ≤ cle 10075   − cmin 10266  ℕ₀cn0 11292  ...cfz 12326  #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303

This theorem is referenced by:  swrdccatid  13497