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Theorem swrdccatin2 13533

Description: The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)

**Hypothesis**
Ref	Expression
swrdccatin12.l	⊢ 𝐿 = (#‘𝐴)

**Assertion**
Ref	Expression
swrdccatin2	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩)))

Proof of Theorem swrdccatin2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdccatin12.l	. . . . . . . 8 ⊢ 𝐿 = (#‘𝐴)
2		oveq1 6697	. . . . . . . . . 10 ⊢ (𝐿 = (#‘𝐴) → (𝐿...𝑁) = ((#‘𝐴)...𝑁))
3	2	eleq2d 2716	. . . . . . . . 9 ⊢ (𝐿 = (#‘𝐴) → (𝑀 ∈ (𝐿...𝑁) ↔ 𝑀 ∈ ((#‘𝐴)...𝑁)))
4		id 22	. . . . . . . . . . 11 ⊢ (𝐿 = (#‘𝐴) → 𝐿 = (#‘𝐴))
5		oveq1 6697	. . . . . . . . . . 11 ⊢ (𝐿 = (#‘𝐴) → (𝐿 + (#‘𝐵)) = ((#‘𝐴) + (#‘𝐵)))
6	4, 5	oveq12d 6708	. . . . . . . . . 10 ⊢ (𝐿 = (#‘𝐴) → (𝐿...(𝐿 + (#‘𝐵))) = ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))))
7	6	eleq2d 2716	. . . . . . . . 9 ⊢ (𝐿 = (#‘𝐴) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ↔ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))))
8	3, 7	anbi12d 747	. . . . . . . 8 ⊢ (𝐿 = (#‘𝐴) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) ↔ (𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))))))
9	1, 8	ax-mp 5	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) ↔ (𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))))
10		lencl 13356	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ₀)
11		elnn0uz 11763	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℕ₀ ↔ (#‘𝐴) ∈ (ℤ_≥‘0))
12	11	biimpi 206	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ (ℤ_≥‘0))
13		fzss1 12418	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘0) → ((#‘𝐴)...𝑁) ⊆ (0...𝑁))
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ ((#‘𝐴) ∈ ℕ₀ → ((#‘𝐴)...𝑁) ⊆ (0...𝑁))
15	14	sseld 3635	. . . . . . . . . 10 ⊢ ((#‘𝐴) ∈ ℕ₀ → (𝑀 ∈ ((#‘𝐴)...𝑁) → 𝑀 ∈ (0...𝑁)))
16		fzss1 12418	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘0) → ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ⊆ (0...((#‘𝐴) + (#‘𝐵))))
17	12, 16	syl 17	. . . . . . . . . . 11 ⊢ ((#‘𝐴) ∈ ℕ₀ → ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ⊆ (0...((#‘𝐴) + (#‘𝐵))))
18	17	sseld 3635	. . . . . . . . . 10 ⊢ ((#‘𝐴) ∈ ℕ₀ → (𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵)))))
19	15, 18	anim12d 585	. . . . . . . . 9 ⊢ ((#‘𝐴) ∈ ℕ₀ → ((𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
20	10, 19	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ ((#‘𝐴)...𝑁) ∧ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
22	9, 21	syl5bi 232	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
23	22	imp 444	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵)))))
24		swrdccatfn 13528	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
25	23, 24	syldan 486	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
26		elfz2 12371	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)))
27		zcn 11420	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
28		zcn 11420	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
29		zcn 11420	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
30	27, 28, 29	3anim123i 1266	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
31	30	3comr 1290	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
32	31	adantr 480	. . . . . . . . . 10 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
33	26, 32	sylbi 207	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
35		nnncan2 10356	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
36	34, 35	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
37	36	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
38	37	oveq2d 6706	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) = (0..^(𝑁 − 𝑀)))
39	38	fneq2d 6020	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀))))
40	25, 39	mpbird 247	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))))
41		simpr 476	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉)
42	41	adantr 480	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐵 ∈ Word 𝑉)
43		elfzmlbm 12488	. . . . 5 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)))
44	43	ad2antrl 764	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)))
45		elfzmlbp 12489	. . . . . . . 8 ⊢ (((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))
46	45	ex 449	. . . . . . 7 ⊢ ((#‘𝐵) ∈ ℤ → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
47		lencl 13356	. . . . . . . . 9 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ₀)
48	47	nn0zd 11518	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
49	48	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘𝐵) ∈ ℤ)
50	46, 49	syl11 33	. . . . . 6 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
51	50	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
52	51	impcom 445	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))
53		swrdvalfn 13472	. . . 4 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)) ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))) → (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩) Fn (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))))
54	42, 44, 52, 53	syl3anc 1366	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩) Fn (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))))
55		simpl 472	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
56	55	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
57		elfzoelz 12509	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ ℤ)
58		elfzelz 12380	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ ℤ)
59		zaddcl 11455	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 + 𝑀) ∈ ℤ)
60	59	expcom 450	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
61	58, 60	syl 17	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
62	61	ad2antrl 764	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
63	57, 62	syl5com 31	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑘 + 𝑀) ∈ ℤ))
64	63	impcom 445	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (𝑘 + 𝑀) ∈ ℤ)
65		df-3an 1056	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑘 + 𝑀) ∈ ℤ))
66	56, 64, 65	sylanbrc 699	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ))
67		ccatsymb 13400	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = if((𝑘 + 𝑀) < (#‘𝐴), (𝐴‘(𝑘 + 𝑀)), (𝐵‘((𝑘 + 𝑀) − (#‘𝐴)))))
68	66, 67	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = if((𝑘 + 𝑀) < (#‘𝐴), (𝐴‘(𝑘 + 𝑀)), (𝐵‘((𝑘 + 𝑀) − (#‘𝐴)))))
69		elfzonn0 12552	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ ℕ₀)
70		zre 11419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
71		zre 11419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
72	70, 71	anim12i 589	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ))
73		elnn0z 11428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ₀ ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
74		zre 11419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
75		0red 10079	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑀 ∈ ℝ → 0 ∈ ℝ)
76	75	anim1i 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
77	76	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
78	77	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
79		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ∈ ℝ)
80	79	anim2i 592	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
81		le2add 10548	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((0 ∈ ℝ ∧ 𝐿 ∈ ℝ) ∧ (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
82	78, 80, 81	syl2anc 694	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
83		recn 10064	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
84	83	addid2d 10275	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐿 ∈ ℝ → (0 + 𝐿) = 𝐿)
85	84	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 + 𝐿) = 𝐿)
86	85	breq1d 4695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 + 𝐿) ≤ (𝑘 + 𝑀) ↔ 𝐿 ≤ (𝑘 + 𝑀)))
87	82, 86	sylibd 229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → 𝐿 ≤ (𝑘 + 𝑀)))
88		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝐿 ∈ ℝ)
89	88	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → 𝐿 ∈ ℝ)
90		readdcl 10057	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 + 𝑀) ∈ ℝ)
91	80, 90	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 + 𝑀) ∈ ℝ)
92	89, 91	lenltd 10221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ (𝑘 + 𝑀) ↔ ¬ (𝑘 + 𝑀) < 𝐿))
93	87, 92	sylibd 229	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → ¬ (𝑘 + 𝑀) < 𝐿))
94	93	expd 451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ≤ 𝑘 → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
95	94	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ≤ 𝑘 → ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
96	95	expd 451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ≤ 𝑘 → (𝑘 ∈ ℝ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))))
97	74, 96	mpan9 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℤ ∧ 0 ≤ 𝑘) → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
98	73, 97	sylbi 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ₀ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
99	72, 98	mpan9 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ₀) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))
100	1	eqcomi 2660	. . . . . . . . . . . . . . . . . . 19 ⊢ (#‘𝐴) = 𝐿
101	100	breq2i 4693	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 + 𝑀) < (#‘𝐴) ↔ (𝑘 + 𝑀) < 𝐿)
102	101	notbii 309	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑘 + 𝑀) < (#‘𝐴) ↔ ¬ (𝑘 + 𝑀) < 𝐿)
103	99, 102	syl6ibr 242	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ₀) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < (#‘𝐴)))
104	103	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ₀ → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < (#‘𝐴))))
105	104	com23 86	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ≤ 𝑀 → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (#‘𝐴))))
106	105	3adant2 1100	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ≤ 𝑀 → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (#‘𝐴))))
107	106	com12 32	. . . . . . . . . . . 12 ⊢ (𝐿 ≤ 𝑀 → ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (#‘𝐴))))
108	107	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁) → ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (#‘𝐴))))
109	108	impcom 445	. . . . . . . . . 10 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (#‘𝐴)))
110	26, 109	sylbi 207	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (#‘𝐴)))
111	110	ad2antrl 764	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (#‘𝐴)))
112	69, 111	syl5com 31	. . . . . . 7 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ¬ (𝑘 + 𝑀) < (#‘𝐴)))
113	112	impcom 445	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → ¬ (𝑘 + 𝑀) < (#‘𝐴))
114	113	iffalsed 4130	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → if((𝑘 + 𝑀) < (#‘𝐴), (𝐴‘(𝑘 + 𝑀)), (𝐵‘((𝑘 + 𝑀) − (#‘𝐴)))) = (𝐵‘((𝑘 + 𝑀) − (#‘𝐴))))
115		zcn 11420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
116	115	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ)
117	28	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ)
118	117	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑀 ∈ ℂ)
119	29	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝐿 ∈ ℂ)
120	116, 118, 119	addsubassd 10450	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀 − 𝐿)))
121		oveq2 6698	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 = (#‘𝐴) → ((𝑘 + 𝑀) − 𝐿) = ((𝑘 + 𝑀) − (#‘𝐴)))
122	121	eqeq1d 2653	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = (#‘𝐴) → (((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀 − 𝐿)) ↔ ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
123	120, 122	syl5ib 234	. . . . . . . . . . . . . 14 ⊢ (𝐿 = (#‘𝐴) → (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
124	1, 123	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿)))
125	124	ex 449	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
126	125	3adant2 1100	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
127	126	adantr 480	. . . . . . . . . 10 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
128	26, 127	sylbi 207	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
129	128	ad2antrl 764	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
130	57, 129	syl5com 31	. . . . . . 7 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
131	130	impcom 445	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → ((𝑘 + 𝑀) − (#‘𝐴)) = (𝑘 + (𝑀 − 𝐿)))
132	131	fveq2d 6233	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (𝐵‘((𝑘 + 𝑀) − (#‘𝐴))) = (𝐵‘(𝑘 + (𝑀 − 𝐿))))
133	68, 114, 132	3eqtrd 2689	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐵‘(𝑘 + (𝑀 − 𝐿))))
134		ccatcl 13392	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
135	134	ad2antrr 762	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
136	1, 12	syl5eqel 2734	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ ℕ₀ → 𝐿 ∈ (ℤ_≥‘0))
137		fzss1 12418	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (ℤ_≥‘0) → (𝐿...𝑁) ⊆ (0...𝑁))
138	10, 136, 137	3syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿...𝑁) ⊆ (0...𝑁))
139	138	sseld 3635	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ (0...𝑁)))
140	139	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ (0...𝑁)))
141	140	com12 32	. . . . . . . 8 ⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑀 ∈ (0...𝑁)))
142	141	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑀 ∈ (0...𝑁)))
143	142	impcom 445	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝑁))
144	143	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → 𝑀 ∈ (0...𝑁))
145	1, 7	ax-mp 5	. . . . . . . . 9 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ↔ 𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))))
146	10, 12	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ (ℤ_≥‘0))
147	146	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘𝐴) ∈ (ℤ_≥‘0))
148	147, 16	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ⊆ (0...((#‘𝐴) + (#‘𝐵))))
149	148	sseld 3635	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵)))))
150	149	impcom 445	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))
151		ccatlen 13393	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
152	151	oveq2d 6706	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (0...(#‘(𝐴 ++ 𝐵))) = (0...((#‘𝐴) + (#‘𝐵))))
153	152	eleq2d 2716	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵)))))
154	153	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵)))))
155	150, 154	mpbird 247	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
156	155	ex 449	. . . . . . . . 9 ⊢ (𝑁 ∈ ((#‘𝐴)...((#‘𝐴) + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
157	145, 156	sylbi 207	. . . . . . . 8 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
158	157	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
159	158	impcom 445	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
160	159	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
161		fzmmmeqm 12412	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
162	161	oveq2d 6706	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) → (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) = (0..^(𝑁 − 𝑀)))
163	162	eleq2d 2716	. . . . . . . 8 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) ↔ 𝑘 ∈ (0..^(𝑁 − 𝑀))))
164	163	biimpd 219	. . . . . . 7 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ (0..^(𝑁 − 𝑀))))
165	164	ad2antrl 764	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ (0..^(𝑁 − 𝑀))))
166	165	imp 444	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → 𝑘 ∈ (0..^(𝑁 − 𝑀)))
167		swrdfv 13469	. . . . 5 ⊢ ((((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)))
168	135, 144, 160, 166, 167	syl31anc 1369	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
169	48, 46	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝑉 → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
170	169	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
171	170	com12 32	. . . . . . . 8 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
172	171	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
173	172	impcom 445	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))
174	42, 44, 173	3jca 1261	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)) ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
175		swrdfv 13469	. . . . 5 ⊢ (((𝐵 ∈ Word 𝑉 ∧ (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)) ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → ((𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩)‘𝑘) = (𝐵‘(𝑘 + (𝑀 − 𝐿))))
176	174, 175	sylan 487	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → ((𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩)‘𝑘) = (𝐵‘(𝑘 + (𝑀 − 𝐿))))
177	133, 168, 176	3eqtr4d 2695	. . 3 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩)‘𝑘))
178	40, 54, 177	eqfnfvd 6354	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩))
179	178	ex 449	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 ifcif 4119 ⟨cop 4216 class class class wbr 4685 Fn wfn 5921 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 + caddc 9977 < clt 10112 ≤ cle 10113 − cmin 10304 ℕ₀cn0 11330 ℤcz 11415 ℤ_≥cuz 11725 ...cfz 12364 ..^cfzo 12504 #chash 13157 Word cword 13323 ++ cconcat 13325 substr csubstr 13327

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-concat 13333 df-substr 13335

This theorem is referenced by: swrdccat3 13538 swrdccatin2d 13546 pfxccat3 41751

W3C validator