swrdccatin2 - Metamath Proof Explorer

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Theorem swrdccatin2 14294

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
swrdccatin2.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		swrdccatin2.l	. . . . . . . 8 ⊢ 𝐿 = (♯‘𝐴)
2		oveq1 7220	. . . . . . . . . 10 ⊢ (𝐿 = (♯‘𝐴) → (𝐿...𝑁) = ((♯‘𝐴)...𝑁))
3	2	eleq2d 2823	. . . . . . . . 9 ⊢ (𝐿 = (♯‘𝐴) → (𝑀 ∈ (𝐿...𝑁) ↔ 𝑀 ∈ ((♯‘𝐴)...𝑁)))
4		id 22	. . . . . . . . . . 11 ⊢ (𝐿 = (♯‘𝐴) → 𝐿 = (♯‘𝐴))
5		oveq1 7220	. . . . . . . . . . 11 ⊢ (𝐿 = (♯‘𝐴) → (𝐿 + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
6	4, 5	oveq12d 7231	. . . . . . . . . 10 ⊢ (𝐿 = (♯‘𝐴) → (𝐿...(𝐿 + (♯‘𝐵))) = ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
7	6	eleq2d 2823	. . . . . . . . 9 ⊢ (𝐿 = (♯‘𝐴) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))
8	3, 7	anbi12d 634	. . . . . . . 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 14088	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
11		elnn0uz 12479	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ₀ ↔ (♯‘𝐴) ∈ (ℤ_≥‘0))
12		fzss1 13151	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ (ℤ_≥‘0) → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
13	11, 12	sylbi 220	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
14	13	sseld 3900	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (𝑀 ∈ ((♯‘𝐴)...𝑁) → 𝑀 ∈ (0...𝑁)))
15		fzss1 13151	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ (ℤ_≥‘0) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
16	11, 15	sylbi 220	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
17	16	sseld 3900	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
18	14, 17	anim12d 612	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
19	10, 18	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
20	19	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
21	9, 20	syl5bi 245	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
22	21	imp 410	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
23		swrdccatfn 14289	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
24	22, 23	syldan 594	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
25		fzmmmeqm 13145	. . . . . . 7 ⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
26	25	oveq2d 7229	. . . . . 6 ⊢ (𝑀 ∈ (𝐿...𝑁) → (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) = (0..^(𝑁 − 𝑀)))
27	26	fneq2d 6473	. . . . 5 ⊢ (𝑀 ∈ (𝐿...𝑁) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀))))
28	27	ad2antrl 728	. . . 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))))
29	24, 28	mpbird 260	. . 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))))
30		simplr 769	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐵 ∈ Word 𝑉)
31		elfzmlbm 13222	. . . . 5 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)))
32	31	ad2antrl 728	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)))
33		lencl 14088	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ₀)
34	33	nn0zd 12280	. . . . . . 7 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
35	34	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘𝐵) ∈ ℤ)
36		elfzmlbp 13223	. . . . . 6 ⊢ (((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
37	35, 36	sylan 583	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
38	37	adantrl 716	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
39		swrdvalfn 14216	. . . 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))))
40	30, 32, 38, 39	syl3anc 1373	. . 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))))
41		simpll 767	. . . . . . 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))
42		elfzelz 13112	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ ℤ)
43		zaddcl 12217	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 + 𝑀) ∈ ℤ)
44	43	expcom 417	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
45	42, 44	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
46	45	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
47		elfzoelz 13243	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ ℤ)
48	46, 47	impel 509	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → (𝑘 + 𝑀) ∈ ℤ)
49		df-3an 1091	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑘 + 𝑀) ∈ ℤ))
50	41, 48, 49	sylanbrc 586	. . . . . 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))
51		ccatsymb 14139	. . . . . 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)))))
52	50, 51	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)))))
53		elfz2 13102	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)))
54		zre 12180	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
55		zre 12180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
56	54, 55	anim12i 616	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ))
57		elnn0z 12189	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ₀ ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
58		zre 12180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
59		0re 10835	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ∈ ℝ
60	59	jctl 527	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
61	60	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
62		id 22	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
63	62	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
64		le2add 11314	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0 ∈ ℝ ∧ 𝐿 ∈ ℝ) ∧ (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
65	61, 63, 64	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
66		recn 10819	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
67	66	addid2d 11033	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (0 + 𝐿) = 𝐿)
68	67	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 + 𝐿) = 𝐿)
69	68	breq1d 5063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 + 𝐿) ≤ (𝑘 + 𝑀) ↔ 𝐿 ≤ (𝑘 + 𝑀)))
70	65, 69	sylibd 242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → 𝐿 ≤ (𝑘 + 𝑀)))
71		simprl 771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → 𝐿 ∈ ℝ)
72		readdcl 10812	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 + 𝑀) ∈ ℝ)
73	72	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 + 𝑀) ∈ ℝ)
74	71, 73	lenltd 10978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ (𝑘 + 𝑀) ↔ ¬ (𝑘 + 𝑀) < 𝐿))
75	70, 74	sylibd 242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → ¬ (𝑘 + 𝑀) < 𝐿))
76	75	expd 419	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ≤ 𝑘 → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
77	76	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ≤ 𝑘 → ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
78	77	expd 419	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ≤ 𝑘 → (𝑘 ∈ ℝ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))))
79	58, 78	mpan9 510	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ 0 ≤ 𝑘) → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
80	57, 79	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
81	56, 80	mpan9 510	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ₀) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))
82	1	breq2i 5061	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 𝑀) < 𝐿 ↔ (𝑘 + 𝑀) < (♯‘𝐴))
83	82	notbii 323	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑘 + 𝑀) < 𝐿 ↔ ¬ (𝑘 + 𝑀) < (♯‘𝐴))
84	81, 83	syl6ib 254	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ₀) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
85	84	ex 416	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ₀ → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
86	85	com23 86	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ≤ 𝑀 → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
87	86	3adant2 1133	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ≤ 𝑀 → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
88	87	imp 410	. . . . . . . . . 10 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝐿 ≤ 𝑀) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
89	88	adantrr 717	. . . . . . . . 9 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
90	53, 89	sylbi 220	. . . . . . . 8 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
91	90	ad2antrl 728	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
92		elfzonn0 13287	. . . . . . 7 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ ℕ₀)
93	91, 92	impel 509	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → ¬ (𝑘 + 𝑀) < (♯‘𝐴))
94	93	iffalsed 4450	. . . . 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))))
95		zcn 12181	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
96	95	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ)
97		zcn 12181	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
98	97	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑀 ∈ ℂ)
99		zcn 12181	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
100	99	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝐿 ∈ ℂ)
101	96, 98, 100	addsubassd 11209	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀 − 𝐿)))
102		oveq2 7221	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = (♯‘𝐴) → ((𝑘 + 𝑀) − 𝐿) = ((𝑘 + 𝑀) − (♯‘𝐴)))
103	102	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝐿 = (♯‘𝐴) → (((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀 − 𝐿)) ↔ ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
104	101, 103	syl5ib 247	. . . . . . . . . . . . 13 ⊢ (𝐿 = (♯‘𝐴) → (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
105	1, 104	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿)))
106	105	ex 416	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
107	106	3adant2 1133	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
108	107	adantr 484	. . . . . . . . 9 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
109	53, 108	sylbi 220	. . . . . . . 8 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
110	109	ad2antrl 728	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
111	110, 47	impel 509	. . . . . 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)))
112	111	fveq2d 6721	. . . . 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))))
113	52, 94, 112	3eqtrd 2781	. . . 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))))
114		ccatcl 14129	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
115	114	adantr 484	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
116	11	biimpi 219	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ (ℤ_≥‘0))
117	1, 116	eqeltrid 2842	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ₀ → 𝐿 ∈ (ℤ_≥‘0))
118		fzss1 13151	. . . . . . . . 9 ⊢ (𝐿 ∈ (ℤ_≥‘0) → (𝐿...𝑁) ⊆ (0...𝑁))
119	10, 117, 118	3syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿...𝑁) ⊆ (0...𝑁))
120	119	sselda 3901	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (𝐿...𝑁)) → 𝑀 ∈ (0...𝑁))
121	120	ad2ant2r 747	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑀 ∈ (0...𝑁))
122	1, 7	ax-mp 5	. . . . . . . . 9 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
123	10, 116, 15	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
124	123	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
125	124	sseld 3900	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
126	125	impcom 411	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))
127		ccatlen 14130	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
128	127	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (0...(♯‘(𝐴 ++ 𝐵))) = (0...((♯‘𝐴) + (♯‘𝐵))))
129	128	eleq2d 2823	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
130	129	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
131	126, 130	mpbird 260	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
132	131	ex 416	. . . . . . . . 9 ⊢ (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
133	122, 132	sylbi 220	. . . . . . . 8 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
134	133	impcom 411	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
135	134	adantrl 716	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
136	115, 121, 135	3jca 1130	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
137	26	eleq2d 2823	. . . . . . 7 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) ↔ 𝑘 ∈ (0..^(𝑁 − 𝑀))))
138	137	ad2antrl 728	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) ↔ 𝑘 ∈ (0..^(𝑁 − 𝑀))))
139	138	biimpa 480	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿)))) → 𝑘 ∈ (0..^(𝑁 − 𝑀)))
140		swrdfv 14213	. . . . 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)))
141	136, 139, 140	syl2an2r 685	. . . 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)))
142	34, 36	sylan 583	. . . . . . 7 ⊢ ((𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
143	142	ad2ant2l 746	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
144	30, 32, 143	3jca 1130	. . . . 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))))
145		swrdfv 14213	. . . . 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))))
146	144, 145	sylan 583	. . . 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))))
147	113, 141, 146	3eqtr4d 2787	. . 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))
148	29, 40, 147	eqfnfvd 6855	. 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)⟩))
149	148	ex 416	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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ifcif 4439 ⟨cop 4547 class class class wbr 5053 Fn wfn 6375 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 + caddc 10732 < clt 10867 ≤ cle 10868 − cmin 11062 ℕ₀cn0 12090 ℤcz 12176 ℤ_≥cuz 12438 ...cfz 13095 ..^cfzo 13238 ♯chash 13896 Word cword 14069 ++ cconcat 14125 substr csubstr 14205

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-concat 14126 df-substr 14206

This theorem is referenced by: pfxccat3 14299 swrdccatin2d 14309

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