Theorem pfxccat3 14767

Description: The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by AV, 10-May-2020.)

Hypothesis

Ref	Expression
swrdccatin2.l	⊢ 𝐿 = (♯‘𝐴)

Assertion

Ref	Expression
pfxccat3	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿)))))))

Proof of Theorem pfxccat3

Step	Hyp	Ref	Expression
1		simpll 778	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simplrl 788	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑀 ∈ (0...𝑁))
3		lencl 14566	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
4		elfznn0 13644	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℕ0)
5	4	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) → 𝑁 ∈ ℕ0)
6	5	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ ℕ0)
7		simplr 780	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → (♯‘𝐴) ∈ ℕ0)
8		swrdccatin2.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (♯‘𝐴)
9	8	breq2i 5118	. . . . . . . . . . . . 13 ⊢ (𝑁 ≤ 𝐿 ↔ 𝑁 ≤ (♯‘𝐴))
10	9	bilani 509	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ≤ (♯‘𝐴))
11		elfz2nn0 13642	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)))
12	6, 7, 10, 11	syl3anbrc 1360	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
13	12	exp31 424	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
14	13	adantl 486	. . . . . . . . 9 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0 → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
15	3, 14	syl5com 32	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
16	15	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
17	16	imp 411	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴))))
18	17	imp 411	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
19	2, 18	jca 520	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
20		swrdccatin1 14758	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
21	1, 19, 20	sylc 66	. . 3 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
22		simp1l 1214	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
23	8	eleq1i 2860	. . . . . . . . . . 11 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
24		elfz2nn0 13642	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁))
25		nn0z 12611	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
26	25	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℤ)
27		nn0z 12611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
28	27	3ad2ant2 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ ℤ)
29	28	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑁 ∈ ℤ)
30		nn0z 12611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ)
31	30	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ)
32	31	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ ℤ)
33	26, 29, 32	3jca 1144	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
34	33	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
35		simpl3 1210	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ≤ 𝑁)
36	35	anim1ci 627	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))
37		elfz2 13538	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)))
38	34, 36, 37	sylanbrc 594	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁))
39	38	exp31 424	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
40	24, 39	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
41	40	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
42	41	com12 33	. . . . . . . . . . 11 ⊢ (𝐿 ∈ ℕ0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
43	23, 42	sylbir 238	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
44	3, 43	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
45	44	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
46	45	imp 411	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))
47	46	a1d 26	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
48	47	3imp 1126	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁))
49		elfz2nn0 13642	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
50		nn0z 12611	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℤ)
51	8, 50	eqeltrid 2873	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ0 → 𝐿 ∈ ℤ)
52	51	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿) → 𝐿 ∈ ℤ)
53	52	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ∈ ℤ)
54		nn0z 12611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 + (♯‘𝐵)) ∈ ℕ0 → (𝐿 + (♯‘𝐵)) ∈ ℤ)
55	54	3ad2ant2 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
56	55	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
57	27	3ad2ant1 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℤ)
58	57	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ ℤ)
59	53, 56, 58	3jca 1144	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
60	8	eqcomi 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (♯‘𝐴) = 𝐿
61	60	eleq1i 2860	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ 𝐿 ∈ ℕ0)
62		nn0re 12509	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
63		nn0re 12509	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
64		ltnle 11285	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
65	62, 63, 64	syl2anr 608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
66	65	bicomd 226	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
67		ltle 11294	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
68	62, 63, 67	syl2anr 608	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
69	66, 68	sylbid 243	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁))
70	69	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)))
71	61, 70	biimtrid 245	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)))
72	71	3ad2ant1 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)))
73	72	imp32 423	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ≤ 𝑁)
74		simpl3 1210	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
75	73, 74	jca 520	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
76		elfz2 13538	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))))
77	59, 75, 76	sylanbrc 594	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
78	77	exp32 425	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
79	49, 78	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
80	79	adantl 486	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
81	3, 80	syl5com 32	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
82	81	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
83	82	imp 411	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
84	83	a1dd 51	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
85	84	3imp 1126	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
86	48, 85	jca 520	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
87	8	swrdccatin2 14762	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩)))
88	22, 86, 87	sylc 66	. . 3 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩))
89		simp1l 1214	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
90		nn0re 12509	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
91	90	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ ℝ)
92		ltnle 11285	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿 ≤ 𝑀))
93	91, 62, 92	syl2anr 608	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 ↔ ¬ 𝐿 ≤ 𝑀))
94	93	bicomd 226	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (¬ 𝐿 ≤ 𝑀 ↔ 𝑀 < 𝐿))
95		simpll 778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ ℕ0)
96		simplr 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈ ℕ0)
97		ltle 11294	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿))
98	90, 62, 97	syl2an 607	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿))
99	98	imp 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ≤ 𝐿)
100		elfz2nn0 13642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿))
101	95, 96, 99, 100	syl3anbrc 1360	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿))
102	101	exp31 424	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿))))
103	102	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿))))
104	103	impcom 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿)))
105	94, 104	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))
106	105	expcom 418	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
107	106	3adant3 1148	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
108	24, 107	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
109	61, 108	biimtrid 245	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
110	109	adantr 485	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
111	3, 110	syl5com 32	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
112	111	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
113	112	imp 411	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))
114	113	a1d 26	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
115	114	3imp 1126	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝑀 ∈ (0...𝐿))
116	63	3ad2ant1 1149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℝ)
117	64	bicomd 226	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
118	62, 116, 117	syl2an 607	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
119	25	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝐿 ∈ ℤ)
120	55	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
121	57	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝑁 ∈ ℤ)
122	119, 120, 121	3jca 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
123	122	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
124	62, 116, 67	syl2an 607	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
125	124	imp 411	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝐿 ≤ 𝑁)
126		simplr3 1234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
127	125, 126	jca 520	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
128	123, 127, 76	sylanbrc 594	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
129	128	ex 417	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
130	118, 129	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
131	130	ex 417	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
132	61, 131	sylbi 220	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
133	3, 132	syl 18	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
134	133	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
135	134	com12 33	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
136	49, 135	sylbi 220	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
137	136	adantl 486	. . . . . . . 8 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
138	137	impcom 412	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
139	138	a1dd 51	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (¬ 𝐿 ≤ 𝑀 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
140	139	3imp 1126	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
141	115, 140	jca 520	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
142	8	pfxccatin12 14766	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿)))))
143	89, 141, 142	sylc 66	. . 3 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿))))
144	21, 88, 143	2if2 4545	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿))))))
145	144	ex 417	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿)))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ifcif 4489  ⟨cop 4597   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  ℝcr 11095  0cc0 11096   + caddc 11099   < clt 11239   ≤ cle 11240   − cmin 11437  ℕ₀cn0 12500  ℤcz 12587  ...cfz 13531  ♯chash 14362  Word cword 14546   ++ cconcat 14603   substr csubstr 14674   prefix cpfx 14704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-n0 12501  df-z 12588  df-uz 12859  df-fz 13532  df-fzo 13679  df-hash 14363  df-word 14547  df-concat 14604  df-substr 14675  df-pfx 14705

This theorem is referenced by:  swrdccat  14768  swrdccat3b  14773