Theorem swrdccatin1 11417

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 6058	. . . . . 6 ⊢ ((♯‘𝐴) = 0 → (0...(♯‘𝐴)) = (0...0))
2	1	eleq2d 2302	. . . . 5 ⊢ ((♯‘𝐴) = 0 → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
3	2	adantl 277	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) = 0) → (𝑁 ∈ (0...(♯‘𝐴)) ↔ 𝑁 ∈ (0...0)))
4		elfz1eq 10369	. . . . . . 7 ⊢ (𝑁 ∈ (0...0) → 𝑁 = 0)
5		elfz1eq 10369	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...0) → 𝑀 = 0)
6		ccatcl 11281	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
7		0z 9588	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
8		swrd00g 11341	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 0 ∈ ℤ) → ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅)
9	6, 7, 8	sylancl 413	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = ∅)
10		simpl 109	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐴 ∈ Word 𝑉)
11		swrd00g 11341	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 0 ∈ ℤ) → (𝐴 substr ⟨0, 0⟩) = ∅)
12	10, 7, 11	sylancl 413	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 substr ⟨0, 0⟩) = ∅)
13	9, 12	eqtr4d 2268	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩))
14	13	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 = 0) → ((𝐴 ++ 𝐵) substr ⟨0, 0⟩) = (𝐴 substr ⟨0, 0⟩))
15		opeq1 3883	. . . . . . . . . . . . . 14 ⊢ (𝑀 = 0 → ⟨𝑀, 0⟩ = ⟨0, 0⟩)
16	15	oveq2d 6066	. . . . . . . . . . . . 13 ⊢ (𝑀 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
17	16	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 = 0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = ((𝐴 ++ 𝐵) substr ⟨0, 0⟩))
18	15	oveq2d 6066	. . . . . . . . . . . . 13 ⊢ (𝑀 = 0 → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
19	18	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 = 0) → (𝐴 substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨0, 0⟩))
20	14, 17, 19	3eqtr4d 2275	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 = 0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
21	5, 20	sylan2 286	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...0)) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩))
22	21	ex 115	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
23	22	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 = 0) → (𝑀 ∈ (0...0) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
24		oveq2 6058	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (0...𝑁) = (0...0))
25	24	eleq2d 2302	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...0)))
26		opeq2 3884	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → ⟨𝑀, 𝑁⟩ = ⟨𝑀, 0⟩)
27	26	oveq2d 6066	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩))
28	26	oveq2d 6066	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝐴 substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 0⟩))
29	27, 28	eqeq12d 2247	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 0⟩) = (𝐴 substr ⟨𝑀, 0⟩)))
30	25, 29	imbi12d 234	. . . . . . . . 9 ⊢ (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⟩))))
31	30	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 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⟩))))
32	23, 31	mpbird 167	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 = 0) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
33	4, 32	sylan2 286	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...0)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
34	33	ex 115	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...0) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
35	34	adantr 276	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) = 0) → (𝑁 ∈ (0...0) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
36	3, 35	sylbid 150	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) = 0) → (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ (0...𝑁) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))))
37	36	impcomd 255	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) = 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
38	6	ad2antrr 488	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
39		simprl 531	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑀 ∈ (0...𝑁))
40		elfzelfzccat 11288	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
41	40	imp 124	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
42	41	ad2ant2rl 511	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
43		swrdvalfn 11348	. . . . 5 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
44	38, 39, 42, 43	syl3anc 1274	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
45		3anass 1009	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ↔ (𝐴 ∈ Word 𝑉 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
46	45	simplbi2 385	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
47	46	ad2antrr 488	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))))
48	47	imp 124	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
49		swrdvalfn 11348	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
50	48, 49	syl 14	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝐴 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
51		simp-4l 543	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐴 ∈ Word 𝑉)
52		simp-4r 544	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝐵 ∈ Word 𝑉)
53		elfznn0 10448	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0)
54		nn0addcl 9531	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
55	54	expcom 116	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ0 → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
56	53, 55	syl 14	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...𝑁) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
57	56	ad2antrl 490	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ ℕ0 → (𝑘 + 𝑀) ∈ ℕ0))
58		elfzonn0 10525	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0)
59	57, 58	impel 280	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
60		lencl 11228	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
61		elnnne0 9510	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐴) ≠ 0))
62	61	simplbi2 385	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
63	60, 62	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
64	63	adantr 276	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ))
65	64	imp 124	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → (♯‘𝐴) ∈ ℕ)
66	65	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (♯‘𝐴) ∈ ℕ)
67		elfzo0 10520	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^(𝑁 − 𝑀)) ↔ (𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)))
68		elfz2nn0 10446	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)))
69		nn0re 9505	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
70	69	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑘 ∈ ℝ)
71		nn0re 9505	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
72	71	ad2antll 491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑀 ∈ ℝ)
73		nn0re 9505	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
74	73	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → 𝑁 ∈ ℝ)
75	70, 72, 74	ltaddsubd 8819	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 ↔ 𝑘 < (𝑁 − 𝑀)))
76		nn0readdcl 9559	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℝ)
77	76	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 + 𝑀) ∈ ℝ)
78		nn0re 9505	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
79	78	ad2antlr 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (♯‘𝐴) ∈ ℝ)
80		ltletr 8363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 + 𝑀) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (♯‘𝐴) ∈ ℝ) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
81	77, 74, 79, 80	syl3anc 1274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (((𝑘 + 𝑀) < 𝑁 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴)))
82	81	expd 258	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((𝑘 + 𝑀) < 𝑁 → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
83	75, 82	sylbird 170	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴))))
84	83	ex 115	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 < (𝑁 − 𝑀) → (𝑁 ≤ (♯‘𝐴) → (𝑘 + 𝑀) < (♯‘𝐴)))))
85	84	com24 87	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (𝑁 ≤ (♯‘𝐴) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (♯‘𝐴)))))
86	85	3impia 1227	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 < (𝑁 − 𝑀) → ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 + 𝑀) < (♯‘𝐴))))
87	86	com13 80	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑘 < (𝑁 − 𝑀) → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
88	87	impancom 260	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
89	88	3adant2 1043	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑀 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑘 + 𝑀) < (♯‘𝐴))))
90	89	com13 80	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
91	68, 90	sylbi 121	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) → (𝑀 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴))))
92	53, 91	mpan9 281	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝑘 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ ∧ 𝑘 < (𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
93	92	adantl 277	. . . . . . . . 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)))
94	67, 93	biimtrid 152	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → (𝑘 ∈ (0..^(𝑁 − 𝑀)) → (𝑘 + 𝑀) < (♯‘𝐴)))
95	94	imp 124	. . . . . . 7 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) < (♯‘𝐴))
96		elfzo0 10520	. . . . . . 7 ⊢ ((𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)) ↔ ((𝑘 + 𝑀) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑘 + 𝑀) < (♯‘𝐴)))
97	59, 66, 95, 96	syl3anbrc 1208	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴)))
98		ccatval1 11285	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐴‘(𝑘 + 𝑀)))
99	51, 52, 97, 98	syl3anc 1274	. . . . 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)))
100	6	ad3antrrr 492	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
101		simplrl 537	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑀 ∈ (0...𝑁))
102	42	adantr 276	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
103		simpr 110	. . . . . 6 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑘 ∈ (0..^(𝑁 − 𝑀)))
104		swrdfv 11345	. . . . . 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)))
105	100, 101, 102, 103, 104	syl31anc 1277	. . . . 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)))
106		swrdfv 11345	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → ((𝐴 substr ⟨𝑀, 𝑁⟩)‘𝑘) = (𝐴‘(𝑘 + 𝑀)))
107	48, 106	sylan 283	. . . . 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)))
108	99, 105, 107	3eqtr4d 2275	. . . 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))
109	44, 50, 108	eqfnfvd 5778	. . 3 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
110	109	ex 115	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐴) ≠ 0) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
111	60	nn0zd 9698	. . . . 5 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℤ)
112		zdceq 9653	. . . . 5 ⊢ (((♯‘𝐴) ∈ ℤ ∧ 0 ∈ ℤ) → DECID (♯‘𝐴) = 0)
113	111, 7, 112	sylancl 413	. . . 4 ⊢ (𝐴 ∈ Word 𝑉 → DECID (♯‘𝐴) = 0)
114		dcne 2423	. . . 4 ⊢ (DECID (♯‘𝐴) = 0 ↔ ((♯‘𝐴) = 0 ∨ (♯‘𝐴) ≠ 0))
115	113, 114	sylib 122	. . 3 ⊢ (𝐴 ∈ Word 𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) ≠ 0))
116	115	adantr 276	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) ≠ 0))
117	37, 110, 116	mpjaodan 806	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 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∅c0 3508  ⟨cop 3692   class class class wbr 4109   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050  ℝcr 8126  0cc0 8127   + caddc 8130   < clt 8308   ≤ cle 8309   − cmin 8444  ℕcn 9237  ℕ₀cn0 9496  ℤcz 9577  ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   ++ cconcat 11278   substr csubstr 11337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279  df-substr 11338

This theorem is referenced by:  pfxccat3  11426  pfxccatpfx1  11428  swrdccatin1d  11435