Theorem swrdccatin1 14636

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 7362	. . . . . 6 ⊢ ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
2	1	eleq2d 2819	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3		elfz1eq 13439	. . . . . 6 ⊢ (𝑁 ∈ (0...0) → 𝑁 = 0)
4		elfz1eq 13439	. . . . . . . 8 ⊢ (𝑀 ∈ (0...0) → 𝑀 = 0)
5		swrd00 14556	. . . . . . . . . 10 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6		swrd00 14556	. . . . . . . . . 10 ⊢ (𝐴 substr ⟨0, 0⟩) = ∅
7	5, 6	eqtr4i 2759	. . . . . . . . 9 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8		opeq1 4826	. . . . . . . . . 10 ⊢ (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
9	8	oveq2d 7370	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
10	8	oveq2d 7370	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
11	7, 9, 10	3eqtr4a 2794	. . . . . . . 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 7362	. . . . . . . . 9 ⊢ (𝑁 = 0 → (0...𝑁) = (0...0))
14	13	eleq2d 2819	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15		opeq2 4827	. . . . . . . . . 10 ⊢ (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
16	15	oveq2d 7370	. . . . . . . . 9 ⊢ (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
17	15	oveq2d 7370	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
18	16, 17	eqeq12d 2749	. . . . . . . 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 14485	. . . . . 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 14491	. . . . . . 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 14563	. . . . 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 14563	. . . . 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 13524	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
42		nn0addcl 12425	. . . . . . . . . . 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 13611	. . . . . . . 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 14444	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
49		elnnne0 12404	. . . . . . . . . . . 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 13604	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)))
56		elfz2nn0 13522	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)))
57		nn0re 12399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
58	57	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
59		nn0re 12399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
60	59	ad2antll 729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
61		nn0re 12399	. . . . . . . . . . . . . . . . . . . . . 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 11726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 ↔ 𝑘 < (𝑁 − 𝑀)))
64		nn0readdcl 12457	. . . . . . . . . . . . . . . . . . . . . . 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 12399	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
67	66	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (♯‘𝐴) ∈ ℝ)
68		ltletr 11214	. . . . . . . . . . . . . . . . . . . . . 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 13604	. . . . . . 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 14488	. . . . . 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 14560	. . . . . 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 14560	. . . . . 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 2778	. . . 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 6975	. . 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 3016	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 1541   ∈ wcel 2113   ≠ wne 2929  ∅c0 4282  ⟨cop 4583   class class class wbr 5095   Fn wfn 6483  ‘cfv 6488  (class class class)co 7354  ℝcr 11014  0cc0 11015   + caddc 11018   < clt 11155   ≤ cle 11156   − cmin 11353  ℕcn 12134  ℕ₀cn0 12390  ...cfz 13411  ..^cfzo 13558  ♯chash 14241  Word cword 14424   ++ cconcat 14481   substr csubstr 14552

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-cnex 11071  ax-resscn 11072  ax-1cn 11073  ax-icn 11074  ax-addcl 11075  ax-addrcl 11076  ax-mulcl 11077  ax-mulrcl 11078  ax-mulcom 11079  ax-addass 11080  ax-mulass 11081  ax-distr 11082  ax-i2m1 11083  ax-1ne0 11084  ax-1rid 11085  ax-rnegex 11086  ax-rrecex 11087  ax-cnre 11088  ax-pre-lttri 11089  ax-pre-lttrn 11090  ax-pre-ltadd 11091  ax-pre-mulgt0 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-er 8630  df-en 8878  df-dom 8879  df-sdom 8880  df-fin 8881  df-card 9841  df-pnf 11157  df-mnf 11158  df-xr 11159  df-ltxr 11160  df-le 11161  df-sub 11355  df-neg 11356  df-nn 12135  df-n0 12391  df-z 12478  df-uz 12741  df-fz 13412  df-fzo 13559  df-hash 14242  df-word 14425  df-concat 14482  df-substr 14553

This theorem is referenced by:  pfxccat3  14645  pfxccatpfx1  14647  swrdccatin1d  14654