Theorem swrdccatin1 14685

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 7371	. . . . . 6 ⊢ ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
2	1	eleq2d 2826	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3		elfz1eq 13487	. . . . . 6 ⊢ (𝑁 ∈ (0...0) → 𝑁 = 0)
4		elfz1eq 13487	. . . . . . . 8 ⊢ (𝑀 ∈ (0...0) → 𝑀 = 0)
5		swrd00 14605	. . . . . . . . . 10 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6		swrd00 14605	. . . . . . . . . 10 ⊢ (𝐴 substr ⟨0, 0⟩) = ∅
7	5, 6	eqtr4i 2766	. . . . . . . . 9 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8		opeq1 4811	. . . . . . . . . 10 ⊢ (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
9	8	oveq2d 7379	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
10	8	oveq2d 7379	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
11	7, 9, 10	3eqtr4a 2801	. . . . . . . 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 7371	. . . . . . . . 9 ⊢ (𝑁 = 0 → (0...𝑁) = (0...0))
14	13	eleq2d 2826	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15		opeq2 4812	. . . . . . . . . 10 ⊢ (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
16	15	oveq2d 7379	. . . . . . . . 9 ⊢ (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
17	15	oveq2d 7379	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
18	16, 17	eqeq12d 2756	. . . . . . . 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 345	. . . . . . 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 259	. . . . . 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 254	. . . 4 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
23	22	impcomd 412	. . 3 ⊢ ((♯‘𝐴) = 0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
24	23	adantl 482	. 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 14534	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
26	25	ad2antrr 732	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
27		simprl 776	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑀 ∈ (0...𝑁))
28		elfzelfzccat 14540	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
29	28	imp 407	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
30	29	ad2ant2rl 755	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
31		swrdvalfn 14612	. . . . 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 1379	. . . 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 1100	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
34	33	simplbi2 501	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
35	34	ad2antrr 732	. . . . . 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 407	. . . . 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 14612	. . . . 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 788	. . . . . 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 789	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐵 ∈ Word 𝑉)
41		elfznn0 13572	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
42		nn0addcl 12470	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
43	42	expcom 414	. . . . . . . . . 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 734	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
46		elfzonn0 13660	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0)
47	45, 46	impel 510	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
48		lencl 14493	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
49		elnnne0 12449	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≠ 0))
50	49	simplbi2 501	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
51	48, 50	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
52	51	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
53	52	imp 407	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → (♯‘𝐴) ∈ ℕ)
54	53	ad2antrr 732	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (♯‘𝐴) ∈ ℕ)
55		elfzo0 13653	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)))
56		elfz2nn0 13570	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)))
57		nn0re 12444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
58	57	ad2antrl 734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
59		nn0re 12444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
60	59	ad2antll 735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
61		nn0re 12444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
62	61	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
63	58, 60, 62	ltaddsubd 11748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 ↔ 𝑘 < (𝑁 − 𝑀)))
64		nn0readdcl 12502	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
65	64	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
66		nn0re 12444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
67	66	ad2antlr 733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (♯‘𝐴) ∈ ℝ)
68		ltletr 11236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (♯‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
69	65, 62, 67, 68	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
70	69	expd 416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
71	63, 70	sylbird 261	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
72	71	ex 413	. . . . . . . . . . . . . . . . . 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 1123	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
77	76	3adant2 1137	. . . . . . . . . . . . 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 218	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
80	41, 79	mpan9 511	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
81	80	adantl 482	. . . . . . . . 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 243	. . . . . . . 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 407	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) < (♯‘𝐴))
84		elfzo0 13653	. . . . . . 7 ⊢ ((𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (♯‘𝐴)))
85	47, 54, 83, 84	syl3anbrc 1350	. . . . . 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 14537	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
87	39, 40, 85, 86	syl3anc 1379	. . . . 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 736	. . . . . 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 782	. . . . . 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 481	. . . . . 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 485	. . . . . 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 14609	. . . . . 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 1381	. . . . 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 14609	. . . . . 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 586	. . . . 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 2785	. . . 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 6981	. . 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 413	. 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 3022	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 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∅c0 4268  ⟨cop 4568   class class class wbr 5079   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363  ℝcr 11035  0cc0 11036   + caddc 11039   < clt 11177   ≤ cle 11178   − cmin 11375  ℕcn 12172  ℕ₀cn0 12435  ...cfz 13459  ..^cfzo 13606  ♯chash 14290  Word cword 14473   ++ cconcat 14530   substr csubstr 14601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607  df-hash 14291  df-word 14474  df-concat 14531  df-substr 14602

This theorem is referenced by:  pfxccat3  14694  pfxccatpfx1  14696  swrdccatin1d  14703