Theorem pczpre 12188

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 9542	. . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
2		eqid 2157	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
3		eqid 2157	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
4	2, 3	pcval 12187	. . 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 402	. 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 521	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℤ)
7	6	zcnd 9293	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℂ)
8	7	div1d 8658	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑁 / 1) = 𝑁)
9	8	eqcomd 2163	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 = (𝑁 / 1))
10		prmuz2 12024	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
11		eqid 2157	. . . . . . . 8 ⊢ 1 = 1
12		eqid 2157	. . . . . . . . 9 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}
13		eqid 2157	. . . . . . . . 9 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )
14	12, 13	pcpre1 12183	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 1 = 1) → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = 0)
15	10, 11, 14	sylancl 410	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = 0)
16	15	adantr 274	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = 0)
17	16	oveq2d 5843	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )) = (𝑆 − 0))
18		eqid 2157	. . . . . . . . . 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 12180	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁))
21	10, 20	sylan 281	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁))
22	21	simpld 111	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℕ0)
23	22	nn0cnd 9151	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℂ)
24	23	subid1d 8180	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − 0) = 𝑆)
25	17, 24	eqtr2d 2191	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))
26		1nn 8850	. . . . 5 ⊢ 1 ∈ ℕ
27		oveq1 5834	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 / 𝑦) = (𝑁 / 𝑦))
28	27	eqeq2d 2169	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑁 / 𝑦)))
29		breq2 3971	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝑁))
30	29	rabbidv 2701	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁})
31	30	supeq1d 6934	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < ))
32	31, 19	eqtr4di 2208	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
33	32	oveq1d 5842	. . . . . . . 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 2169	. . . . . . 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 465	. . . . . 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 5835	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝑁 / 𝑦) = (𝑁 / 1))
37	36	eqeq2d 2169	. . . . . . 7 ⊢ (𝑦 = 1 → (𝑁 = (𝑁 / 𝑦) ↔ 𝑁 = (𝑁 / 1)))
38		breq2 3971	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 1))
39	38	rabbidv 2701	. . . . . . . . . 10 ⊢ (𝑦 = 1 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1})
40	39	supeq1d 6934	. . . . . . . . 9 ⊢ (𝑦 = 1 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ))
41	40	oveq2d 5843	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))
42	41	eqeq2d 2169	. . . . . . 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 465	. . . . . 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 2831	. . . . 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 1307	. . . 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 1218	. . 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 7869	. . . . . 6 ⊢ ℝ ∈ V
48		supex2g 6980	. . . . . 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 2230	. . . 4 ⊢ 𝑆 ∈ V
51	2, 3	pceu 12186	. . . . 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 402	. . . 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 2164	. . . . . . 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 460	. . . . . 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 2482	. . . . 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 5164	. . . 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 411	. . 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 146	. 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 2190	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335  ∃!weu 2006   ∈ wcel 2128   ≠ wne 2327  ∃wrex 2436  {crab 2439  Vcvv 2712   class class class wbr 3967  ℩cio 5136  ‘cfv 5173  (class class class)co 5827  supcsup 6929  ℝcr 7734  0cc0 7735  1c1 7736   < clt 7915   − cmin 8051   / cdiv 8550  ℕcn 8839  2c2 8890  ℕ₀cn0 9096  ℤcz 9173  ℤ_≥cuz 9445  ℚcq 9535  ↑cexp 10428   ∥ cdvds 11695  ℙcprime 12000   pCnt cpc 12175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-isom 5182  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-1o 6366  df-2o 6367  df-er 6483  df-en 6689  df-sup 6931  df-inf 6932  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-fz 9920  df-fzo 10052  df-fl 10179  df-mod 10232  df-seqfrec 10355  df-exp 10429  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-dvds 11696  df-gcd 11843  df-prm 12001  df-pc 12176

This theorem is referenced by:  pczcl  12189  pcmul  12192  pcdiv  12193  pc1  12196  pczdvds  12203  pczndvds  12205  pczndvds2  12207  pcneg  12214