Theorem pfxccat3a 14691

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		simprl 776	. . . . . 6 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		elfznn0 13565	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) → 𝑁 ∈ ℕ0)
3	2	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℕ0)
4	3	adantl 482	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ℕ0)
5		swrdccatin2.l	. . . . . . . . . . 11 ⊢ 𝐿 = (♯‘𝐴)
6		lencl 14486	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
7	5, 6	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℕ0)
8	7	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ ℕ0)
9	8	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℕ0)
10	9	adantl 482	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝐿 ∈ ℕ0)
11		simpl 483	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ≤ 𝐿)
12		elfz2nn0 13563	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑁 ≤ 𝐿))
13	4, 10, 11, 12	syl3anbrc 1350	. . . . . 6 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
14		df-3an 1094	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...𝐿)))
15	1, 13, 14	sylanbrc 589	. . . . 5 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)))
16	5	pfxccatpfx1 14689	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
17	15, 16	syl 17	. . . 4 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
18		iftrue 4460	. . . . 5 ⊢ (𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 prefix 𝑁))
19	18	adantr 481	. . . 4 ⊢ ((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))
20	17, 19	eqtr4d 2777	. . 3 ⊢ ((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)))))
21		simprl 776	. . . . . 6 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
22		elfz2nn0 13563	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
23	5	eleq1i 2830	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
24		nn0ltp1le 12578	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
25		nn0re 12437	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
26		nn0re 12437	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
27		ltnle 11216	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
28	25, 26, 27	syl2an 602	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
29	24, 28	bitr3d 282	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
30	29	3ad2antr1 1195	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
31		simpr3 1203	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ≤ (𝐿 + 𝑀))
32	31	anim1ci 622	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
33		nn0z 12539	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
34	33	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → 𝑁 ∈ ℤ)
35	34	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
36	35	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ℤ)
37		peano2nn0 12468	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
38	37	nn0zd 12540	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℤ)
39	38	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 1) ∈ ℤ)
40	39	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 1) ∈ ℤ)
41		nn0z 12539	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
42	41	3ad2ant2 1140	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (𝐿 + 𝑀) ∈ ℤ)
43	42	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 𝑀) ∈ ℤ)
44	43	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 𝑀) ∈ ℤ)
45		elfz 13458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ (𝐿 + 1) ∈ ℤ ∧ (𝐿 + 𝑀) ∈ ℤ) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀))))
46	36, 40, 44, 45	syl3anc 1379	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀))))
47	32, 46	mpbird 258	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
48	47	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
49	30, 48	sylbird 261	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
50	49	ex 413	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
51	23, 50	sylbir 236	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
52	6, 51	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
53	52	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
54	22, 53	biimtrid 243	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
55	54	imp 407	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
56	55	impcom 408	. . . . . 6 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
57		df-3an 1094	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
58	21, 56, 57	sylanbrc 589	. . . . 5 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
59		pfxccatpfx2.m	. . . . . 6 ⊢ 𝑀 = (♯‘𝐵)
60	5, 59	pfxccatpfx2 14690	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
61	58, 60	syl 17	. . . 4 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
62		iffalse 4463	. . . . 5 ⊢ (¬ 𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
63	62	adantr 481	. . . 4 ⊢ ((¬ 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))))
64	61, 63	eqtr4d 2777	. . 3 ⊢ ((¬ 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)))))
65	20, 64	pm2.61ian 817	. 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)))))
66	65	ex 413	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   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ifcif 4454   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   < clt 11170   ≤ cle 11171   − cmin 11368  ℕ₀cn0 12428  ℤcz 12515  ...cfz 13452  ♯chash 14283  Word cword 14466   ++ cconcat 14523   prefix cpfx 14624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-hash 14284  df-word 14467  df-concat 14524  df-substr 14595  df-pfx 14625

This theorem is referenced by: (None)