Theorem pfxccat3 14264

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 767	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simplrl 777	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑀 ∈ (0...𝑁))
3		lencl 14053	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
4		elfznn0 13170	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℕ0)
5	4	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) → 𝑁 ∈ ℕ0)
6	5	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ ℕ0)
7		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → (♯‘𝐴) ∈ ℕ0)
8		swrdccatin2.l	. . . . . . . . . . . . . . 15 ⊢ 𝐿 = (♯‘𝐴)
9	8	breq2i 5047	. . . . . . . . . . . . . 14 ⊢ (𝑁 ≤ 𝐿 ↔ 𝑁 ≤ (♯‘𝐴))
10	9	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑁 ≤ 𝐿 → 𝑁 ≤ (♯‘𝐴))
11	10	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ≤ (♯‘𝐴))
12		elfz2nn0 13168	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)))
13	6, 7, 11, 12	syl3anbrc 1345	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
14	13	exp31 423	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
15	14	adantl 485	. . . . . . . . 9 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0 → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
16	3, 15	syl5com 31	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
17	16	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴)))))
18	17	imp 410	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝑁 ≤ 𝐿 → 𝑁 ∈ (0...(♯‘𝐴))))
19	18	imp 410	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
20	2, 19	jca 515	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁 ≤ 𝐿) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
21		swrdccatin1 14255	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
22	1, 20, 21	sylc 65	. . 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⟩))
23		simp1l 1199	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
24	8	eleq1i 2821	. . . . . . . . . . 11 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
25		elfz2nn0 13168	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁))
26		nn0z 12165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
27	26	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℤ)
28		nn0z 12165	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
29	28	3ad2ant2 1136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ ℤ)
30	29	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑁 ∈ ℤ)
31		nn0z 12165	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ)
32	31	3ad2ant1 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ)
33	32	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ ℤ)
34	27, 30, 33	3jca 1130	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
35	34	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
36		simpl3 1195	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ≤ 𝑁)
37	36	anim1ci 619	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁))
38		elfz2 13067	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)))
39	35, 37, 38	sylanbrc 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁))
40	39	exp31 423	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
41	25, 40	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
42	41	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℕ0 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
43	42	com12 32	. . . . . . . . . . 11 ⊢ (𝐿 ∈ ℕ0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
44	24, 43	sylbir 238	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
45	3, 44	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
46	45	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
47	46	imp 410	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁)))
48	47	a1d 25	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑀 ∈ (𝐿...𝑁))))
49	48	3imp 1113	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑀 ∈ (𝐿...𝑁))
50		elfz2nn0 13168	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
51		nn0z 12165	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℤ)
52	8, 51	eqeltrid 2835	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ0 → 𝐿 ∈ ℤ)
53	52	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿) → 𝐿 ∈ ℤ)
54	53	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ∈ ℤ)
55		nn0z 12165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 + (♯‘𝐵)) ∈ ℕ0 → (𝐿 + (♯‘𝐵)) ∈ ℤ)
56	55	3ad2ant2 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
57	56	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
58	28	3ad2ant1 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℤ)
59	58	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ ℤ)
60	54, 57, 59	3jca 1130	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
61	8	eqcomi 2745	. . . . . . . . . . . . . . . . . . 19 ⊢ (♯‘𝐴) = 𝐿
62	61	eleq1i 2821	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ 𝐿 ∈ ℕ0)
63		nn0re 12064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
64		nn0re 12064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
65		ltnle 10877	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
66	63, 64, 65	syl2anr 600	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
67	66	bicomd 226	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
68		ltle 10886	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
69	63, 64, 68	syl2anr 600	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
70	67, 69	sylbid 243	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁))
71	70	ex 416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)))
72	62, 71	syl5bi 245	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)))
73	72	3ad2ant1 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝐿 ≤ 𝑁)))
74	73	imp32 422	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝐿 ≤ 𝑁)
75		simpl3 1195	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
76	74, 75	jca 515	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
77		elfz2 13067	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))))
78	60, 76, 77	sylanbrc 586	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁 ≤ 𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
79	78	exp32 424	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
80	50, 79	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
81	80	adantl 485	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
82	3, 81	syl5com 31	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
83	82	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
84	83	imp 410	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
85	84	a1dd 50	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
86	85	3imp 1113	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
87	49, 86	jca 515	. . . 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)))))
88	8	swrdccatin2 14259	. . . 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)⟩)))
89	23, 87, 88	sylc 65	. . 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)⟩))
90		simp1l 1199	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
91		nn0re 12064	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ)
92	91	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ ℝ)
93		ltnle 10877	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿 ≤ 𝑀))
94	92, 63, 93	syl2anr 600	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 ↔ ¬ 𝐿 ≤ 𝑀))
95	94	bicomd 226	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (¬ 𝐿 ≤ 𝑀 ↔ 𝑀 < 𝐿))
96		simpll 767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ ℕ0)
97		simplr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈ ℕ0)
98		ltle 10886	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿))
99	91, 63, 98	syl2an 599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑀 < 𝐿 → 𝑀 ≤ 𝐿))
100	99	imp 410	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ≤ 𝐿)
101		elfz2nn0 13168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿))
102	96, 97, 100, 101	syl3anbrc 1345	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿))
103	102	exp31 423	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿))))
104	103	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿))))
105	104	impcom 411	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 → 𝑀 ∈ (0...𝐿)))
106	95, 105	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))
107	106	expcom 417	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
108	107	3adant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
109	25, 108	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
110	62, 109	syl5bi 245	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
111	110	adantr 484	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
112	3, 111	syl5com 31	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
113	112	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
114	113	imp 410	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿)))
115	114	a1d 25	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (¬ 𝐿 ≤ 𝑀 → 𝑀 ∈ (0...𝐿))))
116	115	3imp 1113	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝑀 ∈ (0...𝐿))
117	64	3ad2ant1 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℝ)
118	65	bicomd 226	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
119	63, 117, 118	syl2an 599	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 ↔ 𝐿 < 𝑁))
120	26	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝐿 ∈ ℤ)
121	56	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
122	58	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝑁 ∈ ℤ)
123	120, 121, 122	3jca 1130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
124	123	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
125	63, 117, 68	syl2an 599	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁 → 𝐿 ≤ 𝑁))
126	125	imp 410	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝐿 ≤ 𝑁)
127		simplr3 1219	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
128	126, 127	jca 515	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))))
129	124, 128, 77	sylanbrc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
130	129	ex 416	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
131	119, 130	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
132	131	ex 416	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
133	62, 132	sylbi 220	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
134	3, 133	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
135	134	adantr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
136	135	com12 32	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
137	50, 136	sylbi 220	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
138	137	adantl 485	. . . . . . . 8 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
139	138	impcom 411	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
140	139	a1dd 50	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (¬ 𝑁 ≤ 𝐿 → (¬ 𝐿 ≤ 𝑀 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
141	140	3imp 1113	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁 ≤ 𝐿 ∧ ¬ 𝐿 ≤ 𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
142	116, 141	jca 515	. . . 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)))))
143	8	pfxccatin12 14263	. . . 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)))))
144	90, 142, 143	sylc 65	. . 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))))
145	22, 89, 144	2if2 4480	. 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))))))
146	145	ex 416	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 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ifcif 4425  ⟨cop 4533   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  ℝcr 10693  0cc0 10694   + caddc 10697   < clt 10832   ≤ cle 10833   − cmin 11027  ℕ₀cn0 12055  ℤcz 12141  ...cfz 13060  ♯chash 13861  Word cword 14034   ++ cconcat 14090   substr csubstr 14170   prefix cpfx 14200

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 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

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 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-card 9520  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-n0 12056  df-z 12142  df-uz 12404  df-fz 13061  df-fzo 13204  df-hash 13862  df-word 14035  df-concat 14091  df-substr 14171  df-pfx 14201

This theorem is referenced by:  swrdccat  14265  swrdccat3b  14270