Theorem pfxccat3 14876

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 779	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simplrl 789	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑀 ∈ (0...𝑁))
3		lencl 14671	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
4		elfznn0 13747	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℕ0)
5	4	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) → 𝑁 ∈ ℕ0)
6	5	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ ℕ0)
7		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → (♯‘𝐴) ∈ ℕ0)
8		swrdccatin2.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (♯‘𝐴)
9	8	breq2i 5111	. . . . . . . . . . . . 13 ⊢ (𝑁 ≤ 𝐿 ↔ 𝑁 ≤ (♯‘𝐴))
10	9	bilani 510	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ≤ (♯‘𝐴))
11		elfz2nn0 13745	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)))
12	6, 7, 10, 11	syl3anbrc 1362	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
13	12	exp31 425	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
14	13	adantl 487	. . . . . . . . 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 486	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
17	16	imp 412	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴))))
18	17	imp 412	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
19	2, 18	jca 521	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
20		swrdccatin1 14867	. . . 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 1216	. . . 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 2852	. . . . . . . . . . 11 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
24		elfz2nn0 13745	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁))
25		nn0z 12710	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
26	25	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℤ)
27		nn0z 12710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
28	27	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ ℤ)
29	28	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑁 ∈ ℤ)
30		nn0z 12710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ)
31	30	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ)
32	31	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ ℤ)
33	26, 29, 32	3jca 1146	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
34	33	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
35		simpl3 1212	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ≤ 𝑁)
36	35	anim1ci 628	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))
37		elfz2 13639	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)))
38	34, 36, 37	sylanbrc 595	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁))
39	38	exp31 425	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
46	45	imp 412	. . . . . . 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 1128	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁))
49		elfz2nn0 13745	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
50		nn0z 12710	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℤ)
51	8, 50	eqeltrid 2865	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ0 → 𝐿 ∈ ℤ)
52	51	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿) → 𝐿 ∈ ℤ)
53	52	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ∈ ℤ)
54		nn0z 12710	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 + (♯‘𝐵)) ∈ ℕ0 → (𝐿 + (♯‘𝐵)) ∈ ℤ)
55	54	3ad2ant2 1152	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
56	55	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
57	27	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℤ)
58	57	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ ℤ)
59	53, 56, 58	3jca 1146	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
60	8	eqcomi 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (♯‘𝐴) = 𝐿
61	60	eleq1i 2852	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ 𝐿 ∈ ℕ0)
62		nn0re 12608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
63		nn0re 12608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
64		ltnle 11382	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
65	62, 63, 64	syl2anr 609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
66	65	bicomd 226	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
67		ltle 11391	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
68	62, 63, 67	syl2anr 609	. . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)))
73	72	imp32 424	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ≤ 𝑁)
74		simpl3 1212	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
75	73, 74	jca 521	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
76		elfz2 13639	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))))
77	59, 75, 76	sylanbrc 595	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
78	77	exp32 426	. . . . . . . . . . . 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 487	. . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
83	82	imp 412	. . . . . . 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 1128	. . . . 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 521	. . . 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 14871	. . . 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 1216	. . . 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 12608	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
91	90	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ ℝ)
92		ltnle 11382	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿 ≤ 𝑀))
93	91, 62, 92	syl2anr 609	. . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ ℕ0)
96		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈ ℕ0)
97		ltle 11391	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿))
98	90, 62, 97	syl2an 608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿))
99	98	imp 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ≤ 𝐿)
100		elfz2nn0 13745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿))
101	95, 96, 99, 100	syl3anbrc 1362	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿))
102	101	exp31 425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿))))
103	102	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿))))
104	103	impcom 413	. . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
107	106	3adant3 1150	. . . . . . . . . . . . 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 486	. . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
113	112	imp 412	. . . . . . 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 1128	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝑀 ∈ (0...𝐿))
116	63	3ad2ant1 1151	. . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
119	25	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝐿 ∈ ℤ)
120	55	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
121	57	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝑁 ∈ ℤ)
122	119, 120, 121	3jca 1146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
123	122	adantr 486	. . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
125	124	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝐿 ≤ 𝑁)
126		simplr3 1236	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
127	125, 126	jca 521	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
128	123, 127, 76	sylanbrc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
129	128	ex 418	. . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . 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 486	. . . . . . . . . . 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 487	. . . . . . . 8 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
138	137	impcom 413	. . . . . . 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 1128	. . . . 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 521	. . . 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 14875	. . . 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 4538	. 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 418	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)))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ifcif 4482  ⟨cop 4590   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  ℝcr 11192  0cc0 11193   + caddc 11196   < clt 11336   ≤ cle 11337   − cmin 11534  ℕ₀cn0 12599  ℤcz 12686  ...cfz 13632  ♯chash 14467  Word cword 14651   ++ cconcat 14708   substr csubstr 14781   prefix cpfx 14813

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-n0 12600  df-z 12687  df-uz 12959  df-fz 13633  df-fzo 13782  df-hash 14468  df-word 14652  df-concat 14709  df-substr 14782  df-pfx 14814

This theorem is used by:  swrdccat  14877  swrdccat3b  14882