Theorem pczpre 12999

Description: Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
pczpre.1	⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < )

Assertion

Ref	Expression
pczpre	⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆)

Distinct variable groups:   𝑛,𝑁   𝑃,𝑛

Allowed substitution hint:   𝑆(𝑛)

Proof of Theorem pczpre

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zq 9961	. . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
2		eqid 2234	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
3		eqid 2234	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
4	2, 3	pcval 12998	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
5	1, 4	sylanr1 404	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
6		simprl 531	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℤ)
7	6	zcnd 9704	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℂ)
8	7	div1d 9056	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑁 / 1) = 𝑁)
9	8	eqcomd 2240	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 = (𝑁 / 1))
10		prmuz2 12832	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
11		eqid 2234	. . . . . . . 8 ⊢ 1 = 1
12		eqid 2234	. . . . . . . . 9 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}
13		eqid 2234	. . . . . . . . 9 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )
14	12, 13	pcpre1 12994	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 1 = 1) → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = 0)
15	10, 11, 14	sylancl 413	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = 0)
16	15	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = 0)
17	16	oveq2d 6068	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )) = (𝑆 − 0))
18		eqid 2234	. . . . . . . . . 10 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}
19		pczpre.1	. . . . . . . . . 10 ⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < )
20	18, 19	pcprecl 12991	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁))
21	10, 20	sylan 283	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁))
22	21	simpld 112	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℕ0)
23	22	nn0cnd 9557	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℂ)
24	23	subid1d 8575	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − 0) = 𝑆)
25	17, 24	eqtr2d 2268	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))
26		1nn 9250	. . . . 5 ⊢ 1 ∈ ℕ
27		oveq1 6059	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 / 𝑦) = (𝑁 / 𝑦))
28	27	eqeq2d 2246	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑁 / 𝑦)))
29		breq2 4115	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝑁))
30	29	rabbidv 2804	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁})
31	30	supeq1d 7280	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < ))
32	31, 19	eqtr4di 2285	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
33	32	oveq1d 6067	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))
34	33	eqeq2d 2246	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
35	28, 34	anbi12d 473	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑁 / 𝑦) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
36		oveq2 6060	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝑁 / 𝑦) = (𝑁 / 1))
37	36	eqeq2d 2246	. . . . . . 7 ⊢ (𝑦 = 1 → (𝑁 = (𝑁 / 𝑦) ↔ 𝑁 = (𝑁 / 1)))
38		breq2 4115	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 1))
39	38	rabbidv 2804	. . . . . . . . . 10 ⊢ (𝑦 = 1 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1})
40	39	supeq1d 7280	. . . . . . . . 9 ⊢ (𝑦 = 1 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ))
41	40	oveq2d 6068	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))
42	41	eqeq2d 2246	. . . . . . 7 ⊢ (𝑦 = 1 → (𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ))))
43	37, 42	anbi12d 473	. . . . . 6 ⊢ (𝑦 = 1 → ((𝑁 = (𝑁 / 𝑦) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))))
44	35, 43	rspc2ev 2938	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℕ ∧ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
45	26, 44	mp3an2 1362	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
46	6, 9, 25, 45	syl12anc 1272	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
47		reex 8263	. . . . . 6 ⊢ ℝ ∈ V
48		supex2g 7326	. . . . . 6 ⊢ (ℝ ∈ V → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < ) ∈ V)
49	47, 48	ax-mp 5	. . . . 5 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < ) ∈ V
50	19, 49	eqeltri 2307	. . . 4 ⊢ 𝑆 ∈ V
51	2, 3	pceu 12997	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
52	1, 51	sylanr1 404	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
53		eqeq1 2241	. . . . . . 7 ⊢ (𝑧 = 𝑆 → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
54	53	anbi2d 464	. . . . . 6 ⊢ (𝑧 = 𝑆 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
55	54	2rexbidv 2569	. . . . 5 ⊢ (𝑧 = 𝑆 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
56	55	iota2 5344	. . . 4 ⊢ ((𝑆 ∈ V ∧ ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆))
57	50, 52, 56	sylancr 414	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆))
58	46, 57	mpbid 147	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆)
59	5, 58	eqtrd 2267	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃!weu 2082   ∈ wcel 2205   ≠ wne 2414  ∃wrex 2523  {crab 2526  Vcvv 2815   class class class wbr 4111  ℩cio 5312  ‘cfv 5354  (class class class)co 6052  supcsup 7275  ℝcr 8128  0cc0 8129  1c1 8130   < clt 8310   − cmin 8446   / cdiv 8948  ℕcn 9239  2c2 9290  ℕ₀cn0 9498  ℤcz 9579  ℤ_≥cuz 9856  ℚcq 9954  ↑cexp 10904   ∥ cdvds 12477  ℙcprime 12808   pCnt cpc 12986

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sup 7277  df-inf 7278  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-fl 10634  df-mod 10689  df-seqfrec 10814  df-exp 10905  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-dvds 12478  df-gcd 12654  df-prm 12809  df-pc 12987

This theorem is referenced by:  pczcl  13000  pcmul  13003  pcdiv  13004  pc1  13007  pczdvds  13016  pczndvds  13018  pczndvds2  13020  pcneg  13027