Theorem swrdccatin1 13334

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 6557	. . . . . . 7 ⊢ ((#‘𝐴) = 0 → (0...(#‘𝐴)) = (0...0))
2	1	eleq2d 2673	. . . . . 6 ⊢ ((#‘𝐴) = 0 → (𝑁 ∈ (0...(#‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3		elfz1eq 12223	. . . . . . 7 ⊢ (𝑁 ∈ (0...0) → 𝑁 = 0)
4		elfz1eq 12223	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...0) → 𝑀 = 0)
5		swrd00 13270	. . . . . . . . . . 11 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅
6		swrd00 13270	. . . . . . . . . . 11 ⊢ (𝐴 substr ⟨0, 0⟩) = ∅
7	5, 6	eqtr4i 2635	. . . . . . . . . 10 ⊢ ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩)
8		opeq1 4340	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
9	8	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
10	8	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
11	7, 9, 10	3eqtr4a 2670	. . . . . . . . 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 6557	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (0...𝑁) = (0...0))
14	13	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
15		opeq2 4341	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
16	15	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
17	15	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
18	16, 17	eqeq12d 2625	. . . . . . . . 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 333	. . . . . . . 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 247	. . . . . . 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 242	. . . . 5 ⊢ ((#‘𝐴) = 0 → (𝑁 ∈ (0...(#‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
23	22	com23 84	. . . 4 ⊢ ((#‘𝐴) = 0 → (𝑀 ∈ (0...𝑁) → (𝑁 ∈ (0...(#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
24	23	impd 446	. . 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 13212	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
27	26	adantl 481	. . . . . . 7 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
28	27	adantr 480	. . . . . 6 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
29		simprl 790	. . . . . 6 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → 𝑀 ∈ (0...𝑁))
30		elfzelfzccat 13217	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(#‘𝐴)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
31	30	adantl 481	. . . . . . . . 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 481	. . . . . . 7 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
34	33	impcom 445	. . . . . 6 ⊢ (((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
35		swrdvalfn 13278	. . . . . 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 1318	. . . . 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 1035	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
38	37	simplbi2 653	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))))
39	38	ad2antrl 760	. . . . . . 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 444	. . . . . 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 13278	. . . . . 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 790	. . . . . . . 8 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝐴 ∈ Word 𝑉)
44	43	ad2antrr 758	. . . . . . 7 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐴 ∈ Word 𝑉)
45		simprr 792	. . . . . . . 8 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
46	45	ad2antrr 758	. . . . . . 7 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐵 ∈ Word 𝑉)
47		elfzo0 12376	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)))
48		elfz2nn0 12300	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁))
49		nn0addcl 11205	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
50	49	expcom 450	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ0 → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
51	50	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
52	48, 51	sylbi 206	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (0...𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
53	52	ad2antrl 760	. . . . . . . . . . . 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 1075	. . . . . . . . . 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 206	. . . . . . . . 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 445	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
58		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
59		df-ne 2782	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ≠ 0 ↔ ¬ (#‘𝐴) = 0)
60		elnnne0 11183	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐴) ∈ ℕ ↔ ((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐴) ≠ 0))
61	60	simplbi2 653	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℕ0 → ((#‘𝐴) ≠ 0 → (#‘𝐴) ∈ ℕ))
62	59, 61	syl5bir 232	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ ℕ0 → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
63	58, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
64	63	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ (#‘𝐴) = 0 → (#‘𝐴) ∈ ℕ))
65	64	impcom 445	. . . . . . . . 9 ⊢ ((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (#‘𝐴) ∈ ℕ)
66	65	ad2antrr 758	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (#‘𝐴) ∈ ℕ)
67		elfz2nn0 12300	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (0...(#‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)))
68		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
69	68	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
70		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
71	70	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → 𝑀 ∈ ℝ)
72	71	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
73		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
74	73	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
75	69, 72, 74	ltaddsubd 10506	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 ↔ 𝑘 < (𝑁 − 𝑀)))
76		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 + 𝑀) ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℝ)
77	49, 76	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
78	77	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
79		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ)
80	79	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (#‘𝐴) ∈ ℝ)
81	80	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (#‘𝐴) ∈ ℝ)
82		ltletr 10008	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴)))
83	78, 74, 81, 82	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴)))
84	83	expd 451	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴))))
85	75, 84	sylbird 249	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴))))
86	85	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (#‘𝐴) → (𝑘 + 𝑀) < (#‘𝐴)))))
87	86	com24 93	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (𝑁 ≤ (#‘𝐴) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (#‘𝐴)))))
88	87	3impia 1253	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (#‘𝐴))))
89	88	com13 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 < (𝑁 − 𝑀) → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
90	89	impancom 455	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
91	90	3adant2 1073	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑘 + 𝑀) < (#‘𝐴))))
92	91	com13 86	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (#‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (#‘𝐴))))
93	67, 92	sylbi 206	. . . . . . . . . . . . . . 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 1075	. . . . . . . . . . . . 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 206	. . . . . . . . . . . 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 448	. . . . . . . . . 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 231	. . . . . . . . 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 444	. . . . . . . 8 ⊢ ((((¬ (#‘𝐴) = 0 ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) < (#‘𝐴))
101		elfzo0 12376	. . . . . . . 8 ⊢ ((𝑘 + 𝑀) ∈ (0..^(#‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (#‘𝐴)))
102	57, 66, 100, 101	syl3anbrc 1239	. . . . . . 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 13214	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(#‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
104	44, 46, 102, 103	syl3anc 1318	. . . . . 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 758	. . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . 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 1235	. . . . . . 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 13276	. . . . . . 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 698	. . . . . 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 13276	. . . . . . 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 487	. . . . . 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 2654	. . . . 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 6222	. . . 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 449	. . 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 449	. 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 175	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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158

This theorem is referenced by:  swrdccat3  13343  swrdccatin1d  13350  pfxccat3  40289  pfxccatpfx1  40290