Theorem pfxccat3a 14773

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 771	. . . . . 6 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		elfznn0 13657	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) → 𝑁 ∈ ℕ0)
3	2	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℕ0)
4	3	adantl 481	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ℕ0)
5		swrdccatin2.l	. . . . . . . . . . 11 ⊢ 𝐿 = (♯‘𝐴)
6		lencl 14568	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
7	5, 6	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℕ0)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ ℕ0)
9	8	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℕ0)
10	9	adantl 481	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝐿 ∈ ℕ0)
11		simpl 482	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ≤ 𝐿)
12		elfz2nn0 13655	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑁 ≤ 𝐿))
13	4, 10, 11, 12	syl3anbrc 1342	. . . . . 6 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
14		df-3an 1088	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...𝐿)))
15	1, 13, 14	sylanbrc 583	. . . . 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 14771	. . . . 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 4537	. . . . 5 ⊢ (𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 prefix 𝑁))
19	18	adantr 480	. . . 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 2778	. . 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 771	. . . . . 6 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
22		elfz2nn0 13655	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
23	5	eleq1i 2830	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
24		nn0ltp1le 12674	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
25		nn0re 12533	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
26		nn0re 12533	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
27		ltnle 11338	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
28	25, 26, 27	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
29	24, 28	bitr3d 281	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
30	29	3ad2antr1 1187	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
31		simpr3 1195	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ≤ (𝐿 + 𝑀))
32	31	anim1ci 616	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
33		nn0z 12636	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
34	33	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → 𝑁 ∈ ℤ)
35	34	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
36	35	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ℤ)
37		peano2nn0 12564	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
38	37	nn0zd 12637	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℤ)
39	38	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 1) ∈ ℤ)
40	39	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 1) ∈ ℤ)
41		nn0z 12636	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
42	41	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (𝐿 + 𝑀) ∈ ℤ)
43	42	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 𝑀) ∈ ℤ)
44	43	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 𝑀) ∈ ℤ)
45		elfz 13550	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
48	47	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
49	30, 48	sylbird 260	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
50	49	ex 412	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
51	23, 50	sylbir 235	. . . . . . . . . . 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 480	. . . . . . . . 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 242	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
55	54	imp 406	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
56	55	impcom 407	. . . . . 6 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
57		df-3an 1088	. . . . . 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 583	. . . . 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 14772	. . . . 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 4540	. . . . 5 ⊢ (¬ 𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
63	62	adantr 480	. . . 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 2778	. . 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 812	. 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 412	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ifcif 4531   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293   ≤ cle 11294   − cmin 11490  ℕ₀cn0 12524  ℤcz 12611  ...cfz 13544  ♯chash 14366  Word cword 14549   ++ cconcat 14605   prefix cpfx 14705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-substr 14676  df-pfx 14706

This theorem is referenced by: (None)