Theorem swrdccatin1 13483

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 6658	. . . . . . 7 ⊢ ((#‘𝐴) = 0 → (0...(#‘𝐴)) = (0...0))
2	1	eleq2d 2687	. . . . . 6 ⊢ ((#‘𝐴) = 0 → (𝑁 ∈ (0...(#‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3		elfz1eq 12352	. . . . . . 7 ⊢ (𝑁 ∈ (0...0) → 𝑁 = 0)
4		elfz1eq 12352	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...0) → 𝑀 = 0)
5		swrd00 13418	. . . . . . . . . . 11 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6		swrd00 13418	. . . . . . . . . . 11 ⊢ (𝐴 substr ⟨0, 0⟩) = ∅
7	5, 6	eqtr4i 2647	. . . . . . . . . 10 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8		opeq1 4402	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
9	8	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
10	8	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
11	7, 9, 10	3eqtr4a 2682	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
12	4, 11	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
13		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (0...𝑁) = (0...0))
14	13	eleq2d 2687	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15		opeq2 4403	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
16	15	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
17	15	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
18	16, 17	eqeq12d 2637	. . . . . . . . 9 ⊢ (𝑁 = 0 → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
19	14, 18	imbi12d 334	. . . . . . . 8 ⊢ (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 248	. . . . . . 7 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
21	3, 20	syl 17	. . . . . 6 ⊢ (𝑁 ∈ (0...0) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
22	2, 21	syl6bi 243	. . . . 5 ⊢ ((#‘𝐴) = 0 → (𝑁 ∈ (0...(#‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
23	22	com23 86	. . . 4 ⊢ ((#‘𝐴) = 0 → (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
24	23	impd 447	. . 3 ⊢ ((#‘𝐴) = 0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
25	24	a1d 25	. 2 ⊢ ((#‘𝐴) = 0 → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
26		ccatcl 13359	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
27	26	adantl 482	. . . . . . 7 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
28	27	adantr 481	. . . . . 6 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
29		simprl 794	. . . . . 6 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → 𝑀 ∈ (0...𝑁))
30		elfzelfzccat 13364	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(#‘𝐴)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
31	30	adantl 482	. . . . . . . . 9 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(#‘𝐴)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
32	31	com12 32	. . . . . . . 8 ⊢ (𝑁 ∈ (0...(#‘𝐴)) → ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
33	32	adantl 482	. . . . . . 7 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
34	33	impcom 446	. . . . . 6 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
35		swrdvalfn 13426	. . . . . 6 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
36	28, 29, 34, 35	syl3anc 1326	. . . . 5 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
37		3anass 1042	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
38	37	simplbi2 655	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
39	38	ad2antrl 764	. . . . . . 7 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
40	39	imp 445	. . . . . 6 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))))
41		swrdvalfn 13426	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
42	40, 41	syl 17	. . . . 5 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
43		simprl 794	. . . . . . . 8 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝐴 ∈ Word 𝑉)
44	43	ad2antrr 762	. . . . . . 7 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐴 ∈ Word 𝑉)
45		simprr 796	. . . . . . . 8 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
46	45	ad2antrr 762	. . . . . . 7 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐵 ∈ Word 𝑉)
47		elfzo0 12508	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)))
48		elfz2nn0 12431	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁))
49		nn0addcl 11328	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
50	49	expcom 451	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ0 → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
51	50	3ad2ant1 1082	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
52	48, 51	sylbi 207	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (0...𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
53	52	ad2antrl 764	. . . . . . . . . . . 12 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
54	53	com12 32	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝑘 + 𝑀) ∈ ℕ0))
55	54	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝑘 + 𝑀) ∈ ℕ0))
56	47, 55	sylbi 207	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) → (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝑘 + 𝑀) ∈ ℕ0))
57	56	impcom 446	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
58		lencl 13324	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
59		df-ne 2795	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ≠ 0 ↔ ¬ (#‘𝐴) = 0)
60		elnnne0 11306	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐴) ∈ ℕ ↔ ((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐴) ≠ 0))
61	60	simplbi2 655	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℕ0 → ((#‘𝐴) ≠ 0 → (#‘𝐴) ∈ ℕ))
62	59, 61	syl5bir 233	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ ℕ0 → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
63	58, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
64	63	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
65	64	impcom 446	. . . . . . . . 9 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (#‘𝐴) ∈ ℕ)
66	65	ad2antrr 762	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (#‘𝐴) ∈ ℕ)
67		elfz2nn0 12431	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (0...(#‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)))
68		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
69	68	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
70		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
71	70	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → 𝑀 ∈ ℝ)
72	71	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
73		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
74	73	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
75	69, 72, 74	ltaddsubd 10627	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 ↔ 𝑘 < (𝑁 − 𝑀)))
76		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 + 𝑀) ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℝ)
77	49, 76	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
78	77	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
79		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ)
80	79	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (#‘𝐴) ∈ ℝ)
81	80	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (#‘𝐴) ∈ ℝ)
82		ltletr 10129	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴)))
83	78, 74, 81, 82	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴)))
84	83	expd 452	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴))))
85	75, 84	sylbird 250	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴))))
86	85	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴)))))
87	86	com24 95	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (𝑁 ≤ (#‘𝐴) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (#‘𝐴)))))
88	87	3impia 1261	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (#‘𝐴))))
89	88	com13 88	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 < (𝑁 − 𝑀) → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
90	89	impancom 456	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
91	90	3adant2 1080	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
92	91	com13 88	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
93	67, 92	sylbi 207	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (0...(#‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
94	93	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ0 → (𝑁 ∈ (0...(#‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
95	94	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 ∈ (0...(#‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
96	48, 95	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(#‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
97	96	a1i 11	. . . . . . . . . . 11 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(#‘𝐴)) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴)))))
98	97	imp32 449	. . . . . . . . . 10 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴)))
99	47, 98	syl5bi 232	. . . . . . . . 9 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝑘 ∈ (0..^(𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴)))
100	99	imp 445	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) < (#‘𝐴))
101		elfzo0 12508	. . . . . . . 8 ⊢ ((𝑘 + 𝑀) ∈ (0..^(#‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (#‘𝐴)))
102	57, 66, 100, 101	syl3anbrc 1246	. . . . . . 7 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ (0..^(#‘𝐴)))
103		ccatval1 13361	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(#‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
104	44, 46, 102, 103	syl3anc 1326	. . . . . 6 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
105	27	ad2antrr 762	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
106	29	adantr 481	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑀 ∈ (0...𝑁))
107	34	adantr 481	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
108	105, 106, 107	3jca 1242	. . . . . . 7 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
109		swrdfv 13424	. . . . . . 7 ⊢ ((((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)))
110	108, 109	sylancom 701	. . . . . 6 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
111		swrdfv 13424	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
112	40, 111	sylan 488	. . . . . 6 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
113	104, 110, 112	3eqtr4d 2666	. . . . 5 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘))
114	36, 42, 113	eqfnfvd 6314	. . . 4 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
115	114	ex 450	. . 3 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
116	115	ex 450	. 2 ⊢ (¬ (#‘𝐴) = 0 → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
117	25, 116	pm2.61i 176	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  ⟨cop 4183   class class class wbr 4653   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ...cfz 12326  ..^cfzo 12465  #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:  swrdccat3  13492  swrdccatin1d  13499  pfxccat3  41426  pfxccatpfx1  41427