Theorem pfxccat3a 11309

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 10339	. . . . . 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 9590	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
4		swrdccatin2.l	. . . . . . . 8 ⊢ 𝐿 = (♯‘𝐴)
5		lencl 11107	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
6	4, 5	eqeltrid 2316	. . . . . . 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 9590	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℤ)
10		zdcle 9546	. . . 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 841	. . . 4 ⊢ (DECID 𝑁 ≤ 𝐿 → (𝑁 ≤ 𝐿 ∨ ¬ 𝑁 ≤ 𝐿))
13		simprl 529	. . . . . . . . 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 10337	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑁 ≤ 𝐿))
18	14, 15, 16, 17	syl3anbrc 1205	. . . . . . . . 9 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
19		df-3an 1004	. . . . . . . . 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 11307	. . . . . . . 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 3608	. . . . . . . 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 2265	. . . . . 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 529	. . . . . . . . 9 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
28		elfz2nn0 10337	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
29	4	eleq1i 2295	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
30		nn0ltp1le 9532	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
31		nn0z 9489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
32		nn0z 9489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
33		zltnle 9515	. . . . . . . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
37		simpr3 1029	. . . . . . . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . . . . . . . . . . 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 9432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
43	42	nn0zd 9590	. . . . . . . . . . . . . . . . . . . . . 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 9489	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
47	46	3ad2ant2 1043	. . . . . . . . . . . . . . . . . . . . . 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 10239	. . . . . . . . . . . . . . . . . . . 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 1271	. . . . . . . . . . . . . . . . . . 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 1004	. . . . . . . . 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 11308	. . . . . . . 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 3611	. . . . . . . 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 2265	. . . . . 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 721	. . . 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 713  DECID wdc 839   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ifcif 3603   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  0cc0 8022  1c1 8023   + caddc 8025   < clt 8204   ≤ cle 8205   − cmin 8340  ℕ₀cn0 9392  ℤcz 9469  ...cfz 10233  ♯chash 11027  Word cword 11103   ++ cconcat 11157   prefix cpfx 11243

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368  df-ihash 11028  df-word 11104  df-concat 11158  df-substr 11217  df-pfx 11244

This theorem is referenced by: (None)