Theorem pfxccat3a 11318

Description: A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by AV, 10-May-2020.)

Hypotheses

Ref	Expression
swrdccatin2.l	⊢ 𝐿 = (♯‘𝐴)
pfxccatpfx2.m	⊢ 𝑀 = (♯‘𝐵)

Assertion

Ref	Expression
pfxccat3a	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))))

Proof of Theorem pfxccat3a

Step	Hyp	Ref	Expression
1		elfznn0 10348	. . . . . 6 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) → 𝑁 ∈ ℕ0)
2	1	adantl 277	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℕ0)
3	2	nn0zd 9599	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
4		swrdccatin2.l	. . . . . . . 8 ⊢ 𝐿 = (♯‘𝐴)
5		lencl 11116	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
6	4, 5	eqeltrid 2318	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℕ0)
7	6	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ ℕ0)
8	7	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℕ0)
9	8	nn0zd 9599	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℤ)
10		zdcle 9555	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐿 ∈ ℤ) → DECID 𝑁 ≤ 𝐿)
11	3, 9, 10	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → DECID 𝑁 ≤ 𝐿)
12		exmiddc 843	. . . 4 ⊢ (DECID 𝑁 ≤ 𝐿 → (𝑁 ≤ 𝐿 ∨ ¬ 𝑁 ≤ 𝐿))
13		simprl 531	. . . . . . . . 9 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
14	2	adantl 277	. . . . . . . . . 10 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ℕ0)
15	8	adantl 277	. . . . . . . . . 10 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝐿 ∈ ℕ0)
16		simpl 109	. . . . . . . . . 10 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ≤ 𝐿)
17		elfz2nn0 10346	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑁 ≤ 𝐿))
18	14, 15, 16, 17	syl3anbrc 1207	. . . . . . . . 9 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
19		df-3an 1006	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...𝐿)))
20	13, 18, 19	sylanbrc 417	. . . . . . . 8 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)))
21	4	pfxccatpfx1 11316	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
22	20, 21	syl 14	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
23		iftrue 3610	. . . . . . . 8 ⊢ (𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 prefix 𝑁))
24	23	adantr 276	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 prefix 𝑁))
25	22, 24	eqtr4d 2267	. . . . . 6 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))))
26	25	ex 115	. . . . 5 ⊢ (𝑁 ≤ 𝐿 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))))
27		simprl 531	. . . . . . . . 9 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
28		elfz2nn0 10346	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
29	4	eleq1i 2297	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
30		nn0ltp1le 9541	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
31		nn0z 9498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
32		nn0z 9498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
33		zltnle 9524	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
34	31, 32, 33	syl2an 289	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
35	30, 34	bitr3d 190	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
36	35	3ad2antr1 1188	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
37		simpr3 1031	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ≤ (𝐿 + 𝑀))
38	37	anim1ci 341	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
39	32	3ad2ant1 1044	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → 𝑁 ∈ ℤ)
40	39	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
41	40	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ℤ)
42		peano2nn0 9441	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
43	42	nn0zd 9599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℤ)
44	43	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 1) ∈ ℤ)
45	44	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 1) ∈ ℤ)
46		nn0z 9498	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
47	46	3ad2ant2 1045	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (𝐿 + 𝑀) ∈ ℤ)
48	47	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 𝑀) ∈ ℤ)
49	48	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 𝑀) ∈ ℤ)
50		elfz 10248	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℤ ∧ (𝐿 + 1) ∈ ℤ ∧ (𝐿 + 𝑀) ∈ ℤ) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀))))
51	41, 45, 49, 50	syl3anc 1273	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀))))
52	38, 51	mpbird 167	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
53	52	ex 115	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
54	36, 53	sylbird 170	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
55	54	ex 115	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
56	29, 55	sylbir 135	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
57	5, 56	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
58	57	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
59	28, 58	biimtrid 152	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
60	59	imp 124	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
61	60	impcom 125	. . . . . . . . 9 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
62		df-3an 1006	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
63	27, 61, 62	sylanbrc 417	. . . . . . . 8 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
64		pfxccatpfx2.m	. . . . . . . . 9 ⊢ 𝑀 = (♯‘𝐵)
65	4, 64	pfxccatpfx2 11317	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
66	63, 65	syl 14	. . . . . . 7 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
67		iffalse 3613	. . . . . . . 8 ⊢ (¬ 𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
68	67	adantr 276	. . . . . . 7 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
69	66, 68	eqtr4d 2267	. . . . . 6 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))))
70	69	ex 115	. . . . 5 ⊢ (¬ 𝑁 ≤ 𝐿 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))))
71	26, 70	jaoi 723	. . . 4 ⊢ ((𝑁 ≤ 𝐿 ∨ ¬ 𝑁 ≤ 𝐿) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))))
72	12, 71	syl 14	. . 3 ⊢ (DECID 𝑁 ≤ 𝐿 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))))
73	11, 72	mpcom 36	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))))
74	73	ex 115	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ifcif 3605   class class class wbr 4088  ‘cfv 5326  (class class class)co 6017  0cc0 8031  1c1 8032   + caddc 8034   < clt 8213   ≤ cle 8214   − cmin 8349  ℕ₀cn0 9401  ℤcz 9478  ...cfz 10242  ♯chash 11036  Word cword 11112   ++ cconcat 11166   prefix cpfx 11252

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-concat 11167  df-substr 11226  df-pfx 11253

This theorem is referenced by: (None)