Theorem pcfac 12302

Description: Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

Assertion

Ref	Expression
pcfac	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))

Distinct variable groups:   𝑃,𝑘   𝑘,𝑁   𝑘,𝑀

Proof of Theorem pcfac

Dummy variables 𝑚 𝑛 𝑥 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5496	. . . . . . . 8 ⊢ (𝑥 = 0 → (ℤ≥‘𝑥) = (ℤ≥‘0))
2		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
3	2	oveq2d 5869	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4		fvoveq1 5876	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(0 / (𝑃↑𝑘))))
5	4	sumeq2sdv 11333	. . . . . . . . 9 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
6	3, 5	eqeq12d 2185	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘)))))
7	1, 6	raleqbidv 2677	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ ∀𝑚 ∈ (ℤ≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘)))))
8	7	imbi2d 229	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))))
9		fveq2 5496	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (ℤ≥‘𝑥) = (ℤ≥‘𝑛))
10		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
11	10	oveq2d 5869	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12		fvoveq1 5876	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑛 / (𝑃↑𝑘))))
13	12	sumeq2sdv 11333	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))
14	11, 13	eqeq12d 2185	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))))
15	9, 14	raleqbidv 2677	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))))
16	15	imbi2d 229	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))))
17		fveq2 5496	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (ℤ≥‘𝑥) = (ℤ≥‘(𝑛 + 1)))
18		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
19	18	oveq2d 5869	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20		fvoveq1 5876	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
21	20	sumeq2sdv 11333	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
22	19, 21	eqeq12d 2185	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘)))))
23	17, 22	raleqbidv 2677	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘)))))
24	23	imbi2d 229	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))))
25		fveq2 5496	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (ℤ≥‘𝑥) = (ℤ≥‘𝑁))
26		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
27	26	oveq2d 5869	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28		fvoveq1 5876	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑁 / (𝑃↑𝑘))))
29	28	sumeq2sdv 11333	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
30	27, 29	eqeq12d 2185	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘)))))
31	25, 30	raleqbidv 2677	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) ↔ ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘)))))
32	31	imbi2d 229	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))))
33		1zzd 9239	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → 1 ∈ ℤ)
34		eluzelz 9496	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘0) → 𝑚 ∈ ℤ)
35	34	adantl 275	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → 𝑚 ∈ ℤ)
36	33, 35	fzfigd 10387	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (1...𝑚) ∈ Fin)
37		isumz 11352	. . . . . . . . . 10 ⊢ (((1 ∈ ℤ ∧ (1...𝑚) ⊆ (ℤ≥‘1) ∧ ∀𝑗 ∈ (ℤ≥‘1)DECID 𝑗 ∈ (1...𝑚)) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
38	37	olcs 731	. . . . . . . . 9 ⊢ ((1...𝑚) ∈ Fin → Σ𝑘 ∈ (1...𝑚)0 = 0)
39	36, 38	syl 14	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → Σ𝑘 ∈ (1...𝑚)0 = 0)
40		0nn0 9150	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
41		elfznn 10010	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
42	41	nnnn0d 9188	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
43		nn0uz 9521	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
44	42, 43	eleqtrdi 2263	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ≥‘0))
45	44	adantl 275	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ (ℤ≥‘0))
46		simpll 524	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
47		pcfaclem 12301	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
48	40, 45, 46, 47	mp3an2i 1337	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
49	48	sumeq2dv 11331	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
50		fac0 10662	. . . . . . . . . . 11 ⊢ (!‘0) = 1
51	50	oveq2i 5864	. . . . . . . . . 10 ⊢ (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
52		pc1 12259	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
53	51, 52	eqtrid 2215	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
54	53	adantr 274	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝑃 pCnt (!‘0)) = 0)
55	39, 49, 54	3eqtr4rd 2214	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
56	55	ralrimiva 2543	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
57		nn0z 9232	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
58	57	adantr 274	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
59		uzid 9501	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
60		peano2uz 9542	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (𝑛 + 1) ∈ (ℤ≥‘𝑛))
61	58, 59, 60	3syl 17	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑛 + 1) ∈ (ℤ≥‘𝑛))
62		uzss 9507	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘𝑛) → (ℤ≥‘(𝑛 + 1)) ⊆ (ℤ≥‘𝑛))
63		ssralv 3211	. . . . . . . . . 10 ⊢ ((ℤ≥‘(𝑛 + 1)) ⊆ (ℤ≥‘𝑛) → (∀𝑚 ∈ (ℤ≥‘𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) → ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))))
64	61, 62, 63	3syl 17	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (∀𝑚 ∈ (ℤ≥‘𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) → ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))))
65		oveq1 5860	. . . . . . . . . . 11 ⊢ ((𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) + (𝑃 pCnt (𝑛 + 1))))
66		simpll 524	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
67		facp1 10664	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
68	66, 67	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
69	68	oveq2d 5869	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
70		simplr 525	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
71		faccl 10669	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
72		nnz 9231	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
73		nnne0 8906	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
74	72, 73	jca 304	. . . . . . . . . . . . . . 15 ⊢ ((!‘𝑛) ∈ ℕ → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
75	66, 71, 74	3syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
76		nn0p1nn 9174	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
77		nnz 9231	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
78		nnne0 8906	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
79	77, 78	jca 304	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 + 1) ∈ ℕ → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
80	66, 76, 79	3syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
81		pcmul 12255	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0) ∧ ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0)) → (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))) = ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))))
82	70, 75, 80, 81	syl3anc 1233	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))) = ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))))
83	69, 82	eqtr2d 2204	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
84	66	adantr 274	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ0)
85	84	nn0zd 9332	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
86		prmnn 12064	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
87	86	ad2antlr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
88		nnexpcl 10489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ)
89	87, 42, 88	syl2an 287	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃↑𝑘) ∈ ℕ)
90		fldivp1 12300	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → ((⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − (⌊‘(𝑛 / (𝑃↑𝑘)))) = if((𝑃↑𝑘) ∥ (𝑛 + 1), 1, 0))
91	85, 89, 90	syl2anc 409	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − (⌊‘(𝑛 / (𝑃↑𝑘)))) = if((𝑃↑𝑘) ∥ (𝑛 + 1), 1, 0))
92		elfzuz 9977	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ≥‘1))
93	66, 76	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ)
94	70, 93	pccld 12254	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
95	94	nn0zd 9332	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
96		elfz5 9973	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ (ℤ≥‘1) ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
97	92, 95, 96	syl2anr 288	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
98		simpllr 529	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
99	84, 76	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℕ)
100	99	nnzd 9333	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
101	42	adantl 275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ0)
102		pcdvdsb 12273	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
103	98, 100, 101, 102	syl3anc 1233	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
104	97, 103	bitr2d 188	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑃↑𝑘) ∥ (𝑛 + 1) ↔ 𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1)))))
105	104	ifbid 3547	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → if((𝑃↑𝑘) ∥ (𝑛 + 1), 1, 0) = if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0))
106	91, 105	eqtrd 2203	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − (⌊‘(𝑛 / (𝑃↑𝑘)))) = if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0))
107	106	sumeq2dv 11331	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)((⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − (⌊‘(𝑛 / (𝑃↑𝑘)))) = Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0))
108		1zzd 9239	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 1 ∈ ℤ)
109		eluzelz 9496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
110	109	adantl 275	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
111	108, 110	fzfigd 10387	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
112		znq 9583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 + 1) ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℚ)
113	100, 89, 112	syl2anc 409	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℚ)
114	113	flqcld 10233	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℤ)
115	114	zcnd 9335	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
116		znq 9583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → (𝑛 / (𝑃↑𝑘)) ∈ ℚ)
117	85, 89, 116	syl2anc 409	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃↑𝑘)) ∈ ℚ)
118	117	flqcld 10233	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℤ)
119	118	zcnd 9335	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
120	111, 115, 119	fsumsub 11415	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)((⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − (⌊‘(𝑛 / (𝑃↑𝑘)))) = (Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))))
121	94	nn0red 9189	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
122	66	nn0red 9189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
123		peano2re 8055	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
124	122, 123	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℝ)
125	110	zred 9334	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
126	93	nnzd 9333	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
127		zdcle 9288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℤ) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
128	95, 126, 127	syl2anc 409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
129		zletric 9256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℤ) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
130	95, 126, 129	syl2anc 409	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
131	130	ord 719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
132	93	nnnn0d 9188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
133		pcdvdsb 12273	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ0) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
134	70, 126, 132, 133	syl3anc 1233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
135	87, 132	nnexpcld 10631	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
136	135	nnzd 9333	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
137		dvdsle 11804	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
138	136, 93, 137	syl2anc 409	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
139	135	nnred 8891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
140	139, 124	lenltd 8037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
141	138, 140	sylibd 148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
142	134, 141	sylbid 149	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
143	131, 142	syld 45	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
144		prmuz2 12085	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
145	144	ad2antlr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ (ℤ≥‘2))
146		bernneq3 10598	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑛 + 1) ∈ ℕ0) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
147	145, 132, 146	syl2anc 409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
148		condc 848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → ((¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))) → ((𝑛 + 1) < (𝑃↑(𝑛 + 1)) → (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))))
149	128, 143, 147, 148	syl3c 63	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
150		eluzle 9499	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
151	150	adantl 275	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
152	121, 124, 125, 149, 151	letrd 8043	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
153		eluz 9500	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
154	95, 110, 153	syl2anc 409	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
155	152, 154	mpbird 166	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))))
156		fzss2 10020	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
157	155, 156	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
158		elfzelz 9981	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℤ)
159	158	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℤ)
160		1zzd 9239	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
161	95	adantr 274	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
162		fzdcel 9996	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
163	159, 160, 161, 162	syl3anc 1233	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
164	163	ralrimiva 2543	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ∀𝑗 ∈ (1...𝑚)DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
165		sumhashdc 12299	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑚) ∈ Fin ∧ (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚) ∧ ∀𝑗 ∈ (1...𝑚)DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1)))) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (♯‘(1...(𝑃 pCnt (𝑛 + 1)))))
166	111, 157, 164, 165	syl3anc 1233	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (♯‘(1...(𝑃 pCnt (𝑛 + 1)))))
167		hashfz1 10717	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 pCnt (𝑛 + 1)) ∈ ℕ0 → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
168	94, 167	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
169	166, 168	eqtrd 2203	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (𝑃 pCnt (𝑛 + 1)))
170	107, 120, 169	3eqtr3d 2211	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))) = (𝑃 pCnt (𝑛 + 1)))
171	111, 115	fsumcl 11363	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
172	111, 119	fsumcl 11363	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
173	94	nn0cnd 9190	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
174	171, 172, 173	subaddd 8248	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))) = (𝑃 pCnt (𝑛 + 1)) ↔ (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) + (𝑃 pCnt (𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘)))))
175	170, 174	mpbid 146	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) + (𝑃 pCnt (𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
176	83, 175	eqeq12d 2185	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) + (𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘)))))
177	65, 176	syl5ib 153	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) → (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘)))))
178	177	ralimdva 2537	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) → ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘)))))
179	64, 178	syld 45	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (∀𝑚 ∈ (ℤ≥‘𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) → ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘)))))
180	179	ex 114	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑃 ∈ ℙ → (∀𝑚 ∈ (ℤ≥‘𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) → ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))))
181	180	a2d 26	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘)))) → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))))
182	8, 16, 24, 32, 56, 181	nn0ind 9326	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘)))))
183	182	imp 123	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
184		oveq2 5861	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
185	184	sumeq1d 11329	. . . . . 6 ⊢ (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
186	185	eqeq2d 2182	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
187	186	rspcv 2830	. . . 4 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → (∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
188	183, 187	syl5 32	. . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
189	188	3impib 1196	. 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
190	189	3com12 1202	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829   ∧ w3a 973   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448   ⊆ wss 3121  ifcif 3526   class class class wbr 3989  ‘cfv 5198  (class class class)co 5853  Fincfn 6718  ℝcr 7773  0cc0 7774  1c1 7775   + caddc 7777   · cmul 7779   < clt 7954   ≤ cle 7955   − cmin 8090   / cdiv 8589  ℕcn 8878  2c2 8929  ℕ₀cn0 9135  ℤcz 9212  ℤ_≥cuz 9487  ℚcq 9578  ...cfz 9965  ⌊cfl 10224  ↑cexp 10475  !cfa 10659  ♯chash 10709  Σcsu 11316   ∥ cdvds 11749  ℙcprime 12061   pCnt cpc 12238

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-2o 6396  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-dvds 11750  df-gcd 11898  df-prm 12062  df-pc 12239

This theorem is referenced by:  pcbc  12303