Theorem swrdccatin1 14697

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 7398	. . . . . 6 ⊢ ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
2	1	eleq2d 2815	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3		elfz1eq 13503	. . . . . 6 ⊢ (𝑁 ∈ (0...0) → 𝑁 = 0)
4		elfz1eq 13503	. . . . . . . 8 ⊢ (𝑀 ∈ (0...0) → 𝑀 = 0)
5		swrd00 14616	. . . . . . . . . 10 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6		swrd00 14616	. . . . . . . . . 10 ⊢ (𝐴 substr ⟨0, 0⟩) = ∅
7	5, 6	eqtr4i 2756	. . . . . . . . 9 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8		opeq1 4840	. . . . . . . . . 10 ⊢ (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
9	8	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
10	8	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
11	7, 9, 10	3eqtr4a 2791	. . . . . . . 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 7398	. . . . . . . . 9 ⊢ (𝑁 = 0 → (0...𝑁) = (0...0))
14	13	eleq2d 2815	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
16	15	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
17	15	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
18	16, 17	eqeq12d 2746	. . . . . . . 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 344	. . . . . . 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 258	. . . . . 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	biimtrdi 253	. . . 4 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
23	22	impcomd 411	. . 3 ⊢ ((♯‘𝐴) = 0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
24	23	adantl 481	. 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 14546	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
26	25	ad2antrr 726	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
27		simprl 770	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑀 ∈ (0...𝑁))
28		elfzelfzccat 14552	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
29	28	imp 406	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
30	29	ad2ant2rl 749	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
31		swrdvalfn 14623	. . . . 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 1373	. . . 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 1094	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
34	33	simplbi2 500	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
35	34	ad2antrr 726	. . . . . 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 406	. . . . 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 14623	. . . . 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 782	. . . . . 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 783	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐵 ∈ Word 𝑉)
41		elfznn0 13588	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
42		nn0addcl 12484	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
43	42	expcom 413	. . . . . . . . . 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 728	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
46		elfzonn0 13675	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0)
47	45, 46	impel 505	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
48		lencl 14505	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
49		elnnne0 12463	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≠ 0))
50	49	simplbi2 500	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
51	48, 50	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
52	51	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
53	52	imp 406	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → (♯‘𝐴) ∈ ℕ)
54	53	ad2antrr 726	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (♯‘𝐴) ∈ ℕ)
55		elfzo0 13668	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)))
56		elfz2nn0 13586	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)))
57		nn0re 12458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
58	57	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
59		nn0re 12458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
60	59	ad2antll 729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
61		nn0re 12458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
62	61	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
63	58, 60, 62	ltaddsubd 11785	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 ↔ 𝑘 < (𝑁 − 𝑀)))
64		nn0readdcl 12516	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
65	64	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
66		nn0re 12458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
67	66	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (♯‘𝐴) ∈ ℝ)
68		ltletr 11273	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (♯‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
69	65, 62, 67, 68	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
70	69	expd 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
71	63, 70	sylbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
72	71	ex 412	. . . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
77	76	3adant2 1131	. . . . . . . . . . . . 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 217	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
80	41, 79	mpan9 506	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
81	80	adantl 481	. . . . . . . . 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	biimtrid 242	. . . . . . . 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 406	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) < (♯‘𝐴))
84		elfzo0 13668	. . . . . . 7 ⊢ ((𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (♯‘𝐴)))
85	47, 54, 83, 84	syl3anbrc 1344	. . . . . 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 14549	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
87	39, 40, 85, 86	syl3anc 1373	. . . . 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	ad3antrrr 730	. . . . . 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 776	. . . . . 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 480	. . . . . 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 484	. . . . . 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 14620	. . . . . 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 1375	. . . . 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 14620	. . . . . 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 580	. . . . 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 2775	. . . 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 7009	. . 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 412	. 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 3013	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∅c0 4299  ⟨cop 4598   class class class wbr 5110   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390  ℝcr 11074  0cc0 11075   + caddc 11078   < clt 11215   ≤ cle 11216   − cmin 11412  ℕcn 12193  ℕ₀cn0 12449  ...cfz 13475  ..^cfzo 13622  ♯chash 14302  Word cword 14485   ++ cconcat 14542   substr csubstr 14612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-concat 14543  df-substr 14613

This theorem is referenced by:  pfxccat3  14706  pfxccatpfx1  14708  swrdccatin1d  14715