Theorem pfxccat3a 14793

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 783	. . . . . 6 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		elfznn0 13661	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) → 𝑁 ∈ ℕ0)
3	2	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℕ0)
4	3	adantl 487	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ℕ0)
5		swrdccatin2.l	. . . . . . . . . . 11 ⊢ 𝐿 = (♯‘𝐴)
6		lencl 14584	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
7	5, 6	eqeltrid 2869	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℕ0)
8	7	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ ℕ0)
9	8	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℕ0)
10	9	adantl 487	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝐿 ∈ ℕ0)
11		simpl 488	. . . . . . 7 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ≤ 𝐿)
12		elfz2nn0 13659	. . . . . . 7 ⊢ (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑁 ≤ 𝐿))
13	4, 10, 11, 12	syl3anbrc 1362	. . . . . 6 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
14		df-3an 1105	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...𝐿)))
15	1, 13, 14	sylanbrc 595	. . . . 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 14791	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
17	15, 16	syl 18	. . . 4 ⊢ ((𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
18		iftrue 4495	. . . . 5 ⊢ (𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 prefix 𝑁))
19	18	adantr 486	. . . 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 2803	. . 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 783	. . . . . 6 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
22		elfz2nn0 13659	. . . . . . . . 9 ⊢ (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
23	5	eleq1i 2856	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
24		nn0ltp1le 12666	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
25		nn0re 12524	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
26		nn0re 12524	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
27		ltnle 11300	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
28	25, 26, 27	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
29	24, 28	bitr3d 284	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
30	29	3ad2antr1 1207	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁 ≤ 𝐿))
31		simpr3 1215	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ≤ (𝐿 + 𝑀))
32	31	anim1ci 628	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → ((𝐿 + 1) ≤ 𝑁 ∧ 𝑁 ≤ (𝐿 + 𝑀)))
33		nn0z 12626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
34	33	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → 𝑁 ∈ ℤ)
35	34	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
36	35	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ℤ)
37		peano2nn0 12555	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
38	37	nn0zd 12627	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℤ)
39	38	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 1) ∈ ℤ)
40	39	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 1) ∈ ℤ)
41		nn0z 12626	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
42	41	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (𝐿 + 𝑀) ∈ ℤ)
43	42	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 𝑀) ∈ ℤ)
44	43	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 𝑀) ∈ ℤ)
45		elfz 13553	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . 15 ⊢ (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
48	47	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
49	30, 48	sylbird 263	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
50	49	ex 418	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
51	23, 50	sylbir 238	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
52	6, 51	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0 ∧ 𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
53	52	adantr 486	. . . . . . . . 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 245	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
55	54	imp 412	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → (¬ 𝑁 ≤ 𝐿 → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
56	55	impcom 413	. . . . . 6 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
57		df-3an 1105	. . . . . 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 595	. . . . 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 14792	. . . . 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 18	. . . 4 ⊢ ((¬ 𝑁 ≤ 𝐿 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
62		iffalse 4498	. . . . 5 ⊢ (¬ 𝑁 ≤ 𝐿 → if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
63	62	adantr 486	. . . 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 2803	. . 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 824	. 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 418	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))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ifcif 4489   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  ℝcr 11110  0cc0 11111  1c1 11112   + caddc 11114   < clt 11254   ≤ cle 11255   − cmin 11452  ℕ₀cn0 12515  ℤcz 12602  ...cfz 13547  ♯chash 14380  Word cword 14564   ++ cconcat 14621   prefix cpfx 14726

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-n0 12516  df-z 12603  df-uz 12875  df-fz 13548  df-fzo 13696  df-hash 14381  df-word 14565  df-concat 14622  df-substr 14695  df-pfx 14727

This theorem is used by: (None)