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Theorem swrdccatin1 14089

Description: The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)

**Assertion**
Ref	Expression
swrdccatin1	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))

Proof of Theorem swrdccatin1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7166	. . . . . 6 ⊢ ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
2	1	eleq2d 2900	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3		elfz1eq 12921	. . . . . 6 ⊢ (𝑁 ∈ (0...0) → 𝑁 = 0)
4		elfz1eq 12921	. . . . . . . 8 ⊢ (𝑀 ∈ (0...0) → 𝑀 = 0)
5		swrd00 14008	. . . . . . . . . 10 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6		swrd00 14008	. . . . . . . . . 10 ⊢ (𝐴 substr ⟨0, 0⟩) = ∅
7	5, 6	eqtr4i 2849	. . . . . . . . 9 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8		opeq1 4805	. . . . . . . . . 10 ⊢ (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
9	8	oveq2d 7174	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
10	8	oveq2d 7174	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
11	7, 9, 10	3eqtr4a 2884	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
12	4, 11	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
13		oveq2 7166	. . . . . . . . 9 ⊢ (𝑁 = 0 → (0...𝑁) = (0...0))
14	13	eleq2d 2900	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15		opeq2 4806	. . . . . . . . . 10 ⊢ (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
16	15	oveq2d 7174	. . . . . . . . 9 ⊢ (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
17	15	oveq2d 7174	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
18	16, 17	eqeq12d 2839	. . . . . . . 8 ⊢ (𝑁 = 0 → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
19	14, 18	imbi12d 347	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)) ↔ (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))))
20	12, 19	mpbiri 260	. . . . . 6 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
21	3, 20	syl 17	. . . . 5 ⊢ (𝑁 ∈ (0...0) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
22	2, 21	syl6bi 255	. . . 4 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
23	22	impcomd 414	. . 3 ⊢ ((♯‘𝐴) = 0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
24	23	adantl 484	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) = 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
25		ccatcl 13928	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
26	25	ad2antrr 724	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
27		simprl 769	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑀 ∈ (0...𝑁))
28		elfzelfzccat 13936	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
29	28	imp 409	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
30	29	ad2ant2rl 747	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
31		swrdvalfn 14015	. . . . 5 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
32	26, 27, 30, 31	syl3anc 1367	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
33		3anass 1091	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
34	33	simplbi2 503	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
35	34	ad2antrr 724	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
36	35	imp 409	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
37		swrdvalfn 14015	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
38	36, 37	syl 17	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
39		simp-4l 781	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐴 ∈ Word 𝑉)
40		simp-4r 782	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐵 ∈ Word 𝑉)
41		elfznn0 13003	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
42		nn0addcl 11935	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → (𝑘 + 𝑀) ∈ ℕ₀)
43	42	expcom 416	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ₀ → (𝑘 ∈ ℕ₀ → (𝑘 + 𝑀) ∈ ℕ₀))
44	41, 43	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...𝑁) → (𝑘 ∈ ℕ₀ → (𝑘 + 𝑀) ∈ ℕ₀))
45	44	ad2antrl 726	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ ℕ₀ → (𝑘 + 𝑀) ∈ ℕ₀))
46		elfzonn0 13085	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) → 𝑘 ∈ ℕ₀)
47	45, 46	impel 508	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ₀)
48		lencl 13885	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
49		elnnne0 11914	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐴) ≠ 0))
50	49	simplbi2 503	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
51	48, 50	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
52	51	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
53	52	imp 409	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → (♯‘𝐴) ∈ ℕ)
54	53	ad2antrr 724	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (♯‘𝐴) ∈ ℕ)
55		elfzo0 13081	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) ↔ (𝑘 ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)))
56		elfz2nn0 13001	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀ ∧ 𝑁 ≤ (♯‘𝐴)))
57		nn0re 11909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
58	57	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → 𝑘 ∈ ℝ)
59		nn0re 11909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
60	59	ad2antll 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → 𝑀 ∈ ℝ)
61		nn0re 11909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
62	61	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → 𝑁 ∈ ℝ)
63	58, 60, 62	ltaddsubd 11242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → ((𝑘 + 𝑀) < 𝑁 ↔ 𝑘 < (𝑁 − 𝑀)))
64		nn0readdcl 11964	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → (𝑘 + 𝑀) ∈ ℝ)
65	64	adantl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → (𝑘 + 𝑀) ∈ ℝ)
66		nn0re 11909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℝ)
67	66	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → (♯‘𝐴) ∈ ℝ)
68		ltletr 10734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (♯‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
69	65, 62, 67, 68	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
70	69	expd 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
71	63, 70	sylbird 262	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀)) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
72	71	ex 415	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴)))))
73	72	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → (𝑘 + 𝑀) < (♯‘𝐴)))))
74	73	3impia 1113	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀ ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → (𝑘 + 𝑀) < (♯‘𝐴))))
75	74	com13 88	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → (𝑘 < (𝑁 − 𝑀) → ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀ ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
76	75	impancom 454	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ₀ → ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀ ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
77	76	3adant2 1127	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ₀ → ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀ ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
78	77	com13 88	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ₀ ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑀 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
79	56, 78	sylbi 219	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
80	41, 79	mpan9 509	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝑘 ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
81	80	adantl 484	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝑘 ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
82	55, 81	syl5bi 244	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ (0..^(𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
83	82	imp 409	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) < (♯‘𝐴))
84		elfzo0 13081	. . . . . . 7 ⊢ ((𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (♯‘𝐴)))
85	47, 54, 83, 84	syl3anbrc 1339	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)))
86		ccatval1 13932	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
87	39, 40, 85, 86	syl3anc 1367	. . . . 5 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
88	25	ad5ant12 754	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
89		simplrl 775	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑀 ∈ (0...𝑁))
90	30	adantr 483	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
91		simpr 487	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑘 ∈ (0..^(𝑁 − 𝑀)))
92		swrdfv 14012	. . . . . 6 ⊢ ((((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
93	88, 89, 90, 91, 92	syl31anc 1369	. . . . 5 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
94		swrdfv 14012	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
95	36, 94	sylan 582	. . . . 5 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
96	87, 93, 95	3eqtr4d 2868	. . . 4 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘))
97	32, 38, 96	eqfnfvd 6807	. . 3 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
98	97	ex 415	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
99	24, 98	pm2.61dane 3106	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 ⟨cop 4575 class class class wbr 5068 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 ℕ₀cn0 11900 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 ++ cconcat 13924 substr csubstr 14004

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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-substr 14005

This theorem is referenced by: pfxccat3 14098 pfxccatpfx1 14100 swrdccatin1d 14107

W3C validator