swrdccatin2 - Intuitionistic Logic Explorer

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Theorem swrdccatin2 11220

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 5974	. . . . . . . . . 10 ⊢ (𝐿 = (♯‘𝐴) → (𝐿...𝑁) = ((♯‘𝐴)...𝑁))
3	2	eleq2d 2277	. . . . . . . . 9 ⊢ (𝐿 = (♯‘𝐴) → (𝑀 ∈ (𝐿...𝑁) ↔ 𝑀 ∈ ((♯‘𝐴)...𝑁)))
4		id 19	. . . . . . . . . . 11 ⊢ (𝐿 = (♯‘𝐴) → 𝐿 = (♯‘𝐴))
5		oveq1 5974	. . . . . . . . . . 11 ⊢ (𝐿 = (♯‘𝐴) → (𝐿 + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
6	4, 5	oveq12d 5985	. . . . . . . . . 10 ⊢ (𝐿 = (♯‘𝐴) → (𝐿...(𝐿 + (♯‘𝐵))) = ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
7	6	eleq2d 2277	. . . . . . . . 9 ⊢ (𝐿 = (♯‘𝐴) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))
8	3, 7	anbi12d 473	. . . . . . . 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 11035	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
11		elnn0uz 9721	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ₀ ↔ (♯‘𝐴) ∈ (ℤ_≥‘0))
12		fzss1 10220	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ (ℤ_≥‘0) → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
13	11, 12	sylbi 121	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
14	13	sseld 3200	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (𝑀 ∈ ((♯‘𝐴)...𝑁) → 𝑀 ∈ (0...𝑁)))
15		fzss1 10220	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ (ℤ_≥‘0) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
16	11, 15	sylbi 121	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
17	16	sseld 3200	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
18	14, 17	anim12d 335	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
19	10, 18	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
20	19	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
21	9, 20	biimtrid 152	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
22	21	imp 124	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
23		swrdccatfn 11215	. . . . 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 282	. . . 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 10215	. . . . . . 7 ⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
26	25	oveq2d 5983	. . . . . 6 ⊢ (𝑀 ∈ (𝐿...𝑁) → (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) = (0..^(𝑁 − 𝑀)))
27	26	fneq2d 5384	. . . . 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 490	. . . 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 167	. . 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 528	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐵 ∈ Word 𝑉)
31		elfzmlbm 10288	. . . . 5 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)))
32	31	ad2antrl 490	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀 − 𝐿) ∈ (0...(𝑁 − 𝐿)))
33		lencl 11035	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ₀)
34	33	nn0zd 9528	. . . . . . 7 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
35	34	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘𝐵) ∈ ℤ)
36		elfzmlbp 10289	. . . . . 6 ⊢ (((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
37	35, 36	sylan 283	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
38	37	adantrl 478	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
39		swrdvalfn 11147	. . . 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 1250	. . 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 527	. . . . . . 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 10182	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ ℤ)
43		zaddcl 9447	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 + 𝑀) ∈ ℤ)
44	43	expcom 116	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
45	42, 44	syl 14	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
46	45	ad2antrl 490	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
47		elfzoelz 10304	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ ℤ)
48	46, 47	impel 280	. . . . . . 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 983	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑘 + 𝑀) ∈ ℤ))
50	41, 48, 49	sylanbrc 417	. . . . . 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 11096	. . . . . 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 14	. . . . 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 10172	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)))
54		zre 9411	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
55		zre 9411	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
56	54, 55	anim12i 338	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ))
57		elnn0z 9420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ₀ ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
58		zre 9411	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
59		0re 8107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ∈ ℝ
60	59	jctl 314	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
61	60	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
62		id 19	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
63	62	adantrl 478	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
64		le2add 8552	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0 ∈ ℝ ∧ 𝐿 ∈ ℝ) ∧ (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
65	61, 63, 64	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
66		recn 8093	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
67	66	addlidd 8257	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (0 + 𝐿) = 𝐿)
68	67	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 + 𝐿) = 𝐿)
69	68	breq1d 4069	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 + 𝐿) ≤ (𝑘 + 𝑀) ↔ 𝐿 ≤ (𝑘 + 𝑀)))
70	65, 69	sylibd 149	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → 𝐿 ≤ (𝑘 + 𝑀)))
71		simprl 529	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → 𝐿 ∈ ℝ)
72		readdcl 8086	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 + 𝑀) ∈ ℝ)
73	72	adantrl 478	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 + 𝑀) ∈ ℝ)
74	71, 73	lenltd 8225	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ (𝑘 + 𝑀) ↔ ¬ (𝑘 + 𝑀) < 𝐿))
75	70, 74	sylibd 149	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘 ∧ 𝐿 ≤ 𝑀) → ¬ (𝑘 + 𝑀) < 𝐿))
76	75	expd 258	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ≤ 𝑘 → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
77	76	com12 30	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ≤ 𝑘 → ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
78	77	expd 258	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ≤ 𝑘 → (𝑘 ∈ ℝ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))))
79	58, 78	mpan9 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℤ ∧ 0 ≤ 𝑘) → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
80	57, 79	sylbi 121	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
81	56, 80	mpan9 281	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ₀) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))
82	1	breq2i 4067	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 𝑀) < 𝐿 ↔ (𝑘 + 𝑀) < (♯‘𝐴))
83	82	notbii 670	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑘 + 𝑀) < 𝐿 ↔ ¬ (𝑘 + 𝑀) < (♯‘𝐴))
84	81, 83	imbitrdi 161	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ₀) → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
85	84	ex 115	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ₀ → (𝐿 ≤ 𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
86	85	com23 78	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ≤ 𝑀 → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
87	86	3adant2 1019	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ≤ 𝑀 → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
88	87	imp 124	. . . . . . . . . 10 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝐿 ≤ 𝑀) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
89	88	adantrr 479	. . . . . . . . 9 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
90	53, 89	sylbi 121	. . . . . . . 8 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
91	90	ad2antrl 490	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℕ₀ → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
92		elfzonn0 10347	. . . . . . 7 ⊢ (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) → 𝑘 ∈ ℕ₀)
93	91, 92	impel 280	. . . . . 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 3589	. . . . 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 9412	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
96	95	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ)
97		zcn 9412	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
98	97	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑀 ∈ ℂ)
99		zcn 9412	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
100	99	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝐿 ∈ ℂ)
101	96, 98, 100	addsubassd 8438	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀 − 𝐿)))
102		oveq2 5975	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = (♯‘𝐴) → ((𝑘 + 𝑀) − 𝐿) = ((𝑘 + 𝑀) − (♯‘𝐴)))
103	102	eqeq1d 2216	. . . . . . . . . . . . . 14 ⊢ (𝐿 = (♯‘𝐴) → (((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀 − 𝐿)) ↔ ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
104	101, 103	imbitrid 154	. . . . . . . . . . . . 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 115	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
107	106	3adant2 1019	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
108	107	adantr 276	. . . . . . . . 9 ⊢ (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
109	53, 108	sylbi 121	. . . . . . . 8 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀 − 𝐿))))
110	109	ad2antrl 490	. . . . . . 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 280	. . . . . 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 5603	. . . . 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 2244	. . . 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 11087	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
115	114	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
116	11	biimpi 120	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ (ℤ_≥‘0))
117	1, 116	eqeltrid 2294	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ₀ → 𝐿 ∈ (ℤ_≥‘0))
118		fzss1 10220	. . . . . . . . 9 ⊢ (𝐿 ∈ (ℤ_≥‘0) → (𝐿...𝑁) ⊆ (0...𝑁))
119	10, 117, 118	3syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → (𝐿...𝑁) ⊆ (0...𝑁))
120	119	sselda 3201	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (𝐿...𝑁)) → 𝑀 ∈ (0...𝑁))
121	120	ad2ant2r 509	. . . . . 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 17	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
124	123	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
125	124	sseld 3200	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
126	125	impcom 125	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))
127		ccatlen 11089	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
128	127	oveq2d 5983	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (0...(♯‘(𝐴 ++ 𝐵))) = (0...((♯‘𝐴) + (♯‘𝐵))))
129	128	eleq2d 2277	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
130	129	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
131	126, 130	mpbird 167	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
132	131	ex 115	. . . . . . . . 9 ⊢ (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
133	122, 132	sylbi 121	. . . . . . . 8 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
134	133	impcom 125	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
135	134	adantrl 478	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
136	115, 121, 135	3jca 1180	. . . . 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 2277	. . . . . . 7 ⊢ (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁 − 𝐿) − (𝑀 − 𝐿))) ↔ 𝑘 ∈ (0..^(𝑁 − 𝑀))))
138	137	ad2antrl 490	. . . . . 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 296	. . . . 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 11144	. . . . 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 595	. . . 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 283	. . . . . . 7 ⊢ ((𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
143	142	ad2ant2l 508	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(♯‘𝐵)))
144	30, 32, 143	3jca 1180	. . . . 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 11144	. . . . 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 283	. . . 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 2250	. . 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 5703	. 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 115	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 set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 981 = wceq 1373 ∈ wcel 2178 ⊆ wss 3174 ifcif 3579 ⟨cop 3646 class class class wbr 4059 Fn wfn 5285 ‘cfv 5290 (class class class)co 5967 ℂcc 7958 ℝcr 7959 0cc0 7960 + caddc 7963 < clt 8142 ≤ cle 8143 − cmin 8278 ℕ₀cn0 9330 ℤcz 9407 ℤ_≥cuz 9683 ...cfz 10165 ..^cfzo 10299 ♯chash 10957 Word cword 11031 ++ cconcat 11084 substr csubstr 11136

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-addcom 8060 ax-addass 8062 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-0id 8068 ax-rnegex 8069 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-frec 6500 df-1o 6525 df-er 6643 df-en 6851 df-dom 6852 df-fin 6853 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-inn 9072 df-n0 9331 df-z 9408 df-uz 9684 df-fz 10166 df-fzo 10300 df-ihash 10958 df-word 11032 df-concat 11085 df-substr 11137

This theorem is referenced by: pfxccat3 11225 swrdccatin2d 11235

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