Theorem pcfac 16839

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = 0 → (ℤ≥‘𝑥) = (ℤ≥‘0))
2		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
3	2	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4		fvoveq1 7391	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(0 / (𝑃↑𝑘))))
5	4	sumeq2sdv 15638	. . . . . . . . 9 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
6	3, 5	eqeq12d 2753	. . . . . . . 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 3318	. . . . . . 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 340	. . . . . 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 6842	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (ℤ≥‘𝑥) = (ℤ≥‘𝑛))
10		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
11	10	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12		fvoveq1 7391	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑛 / (𝑃↑𝑘))))
13	12	sumeq2sdv 15638	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))
14	11, 13	eqeq12d 2753	. . . . . . . 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 3318	. . . . . . 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 340	. . . . . 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 6842	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (ℤ≥‘𝑥) = (ℤ≥‘(𝑛 + 1)))
18		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
19	18	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20		fvoveq1 7391	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
21	20	sumeq2sdv 15638	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
22	19, 21	eqeq12d 2753	. . . . . . . 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 3318	. . . . . . 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 340	. . . . . 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 6842	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (ℤ≥‘𝑥) = (ℤ≥‘𝑁))
26		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
27	26	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28		fvoveq1 7391	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑁 / (𝑃↑𝑘))))
29	28	sumeq2sdv 15638	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
30	27, 29	eqeq12d 2753	. . . . . . . 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 3318	. . . . . . 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 340	. . . . . 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		fzfid 13908	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (1...𝑚) ∈ Fin)
34		sumz 15657	. . . . . . . . . 10 ⊢ (((1...𝑚) ⊆ (ℤ≥‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
35	34	olcs 877	. . . . . . . . 9 ⊢ ((1...𝑚) ∈ Fin → Σ𝑘 ∈ (1...𝑚)0 = 0)
36	33, 35	syl 17	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → Σ𝑘 ∈ (1...𝑚)0 = 0)
37		0nn0 12428	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
38		elfznn 13481	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
39	38	nnnn0d 12474	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
40		nn0uz 12801	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
41	39, 40	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ≥‘0))
42	41	adantl 481	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ (ℤ≥‘0))
43		simpll 767	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
44		pcfaclem 16838	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
45	37, 42, 43, 44	mp3an2i 1469	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
46	45	sumeq2dv 15637	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
47		fac0 14211	. . . . . . . . . . 11 ⊢ (!‘0) = 1
48	47	oveq2i 7379	. . . . . . . . . 10 ⊢ (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
49		pc1 16795	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
50	48, 49	eqtrid 2784	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
51	50	adantr 480	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝑃 pCnt (!‘0)) = 0)
52	36, 46, 51	3eqtr4rd 2783	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
53	52	ralrimiva 3130	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
54		nn0z 12524	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
55	54	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
56		uzid 12778	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
57		peano2uz 12826	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (𝑛 + 1) ∈ (ℤ≥‘𝑛))
58	55, 56, 57	3syl 18	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑛 + 1) ∈ (ℤ≥‘𝑛))
59		uzss 12786	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘𝑛) → (ℤ≥‘(𝑛 + 1)) ⊆ (ℤ≥‘𝑛))
60		ssralv 4004	. . . . . . . . . 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)))))
61	58, 59, 60	3syl 18	. . . . . . . . 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)))))
62		oveq1 7375	. . . . . . . . . . 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))))
63		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
64		facp1 14213	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
65	63, 64	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
66	65	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
67		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
68		faccl 14218	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
69		nnz 12521	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
70		nnne0 12191	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
71	69, 70	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((!‘𝑛) ∈ ℕ → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
72	63, 68, 71	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
73		nn0p1nn 12452	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
74		nnz 12521	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
75		nnne0 12191	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
76	74, 75	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 + 1) ∈ ℕ → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
77	63, 73, 76	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
78		pcmul 16791	. . . . . . . . . . . . . 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))))
79	67, 72, 77, 78	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))) = ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))))
80	66, 79	eqtr2d 2773	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
81	63	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ0)
82	81	nn0zd 12525	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
83		prmnn 16613	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
84	83	ad2antlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
85		nnexpcl 14009	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ)
86	84, 39, 85	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃↑𝑘) ∈ ℕ)
87		fldivp1 16837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → ((⌊‘((𝑛 + 1) / (𝑃↑𝑘))) − (⌊‘(𝑛 / (𝑃↑𝑘)))) = if((𝑃↑𝑘) ∥ (𝑛 + 1), 1, 0))
88	82, 86, 87	syl2anc 585	. . . . . . . . . . . . . . . 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))
89		elfzuz 13448	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ≥‘1))
90	63, 73	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ)
91	67, 90	pccld 16790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
92	91	nn0zd 12525	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
93		elfz5 13444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ (ℤ≥‘1) ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
94	89, 92, 93	syl2anr 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
95		simpllr 776	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
96	81, 73	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℕ)
97	96	nnzd 12526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
98	39	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ0)
99		pcdvdsb 16809	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
100	95, 97, 98, 99	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
101	94, 100	bitr2d 280	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑃↑𝑘) ∥ (𝑛 + 1) ↔ 𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1)))))
102	101	ifbid 4505	. . . . . . . . . . . . . . . 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))
103	88, 102	eqtrd 2772	. . . . . . . . . . . . . . 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))
104	103	sumeq2dv 15637	. . . . . . . . . . . . . 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))
105		fzfid 13908	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
106	63	nn0red 12475	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
107		peano2re 11318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
108	106, 107	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℝ)
109	108	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℝ)
110	109, 86	nndivred 12211	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℝ)
111	110	flcld 13730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℤ)
112	111	zcnd 12609	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
113	106	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℝ)
114	113, 86	nndivred 12211	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃↑𝑘)) ∈ ℝ)
115	114	flcld 13730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℤ)
116	115	zcnd 12609	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
117	105, 112, 116	fsumsub 15723	. . . . . . . . . . . . . 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)))))
118		fzfi 13907	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ∈ Fin
119	91	nn0red 12475	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
120		eluzelz 12773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
121	120	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
122	121	zred 12608	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
123		prmuz2 16635	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
124	123	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ (ℤ≥‘2))
125	90	nnnn0d 12474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
126		bernneq3 14166	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑛 + 1) ∈ ℕ0) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
127	124, 125, 126	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
128	119, 108	letrid 11297	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
129	128	ord 865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
130	90	nnzd 12526	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
131		pcdvdsb 16809	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ0) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
132	67, 130, 125, 131	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
133	84, 125	nnexpcld 14180	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
134	133	nnzd 12526	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
135		dvdsle 16249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
136	134, 90, 135	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
137	133	nnred 12172	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
138	137, 108	lenltd 11291	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
139	136, 138	sylibd 239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
140	132, 139	sylbid 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
141	129, 140	syld 47	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
142	127, 141	mt4d 117	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
143		eluzle 12776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
144	143	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
145	119, 108, 122, 142, 144	letrd 11302	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
146		eluz 12777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
147	92, 121, 146	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
148	145, 147	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))))
149		fzss2 13492	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
150	148, 149	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
151		sumhash 16836	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑚) ∈ Fin ∧ (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚)) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (♯‘(1...(𝑃 pCnt (𝑛 + 1)))))
152	118, 150, 151	sylancr 588	. . . . . . . . . . . . . . 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)))))
153		hashfz1 14281	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 pCnt (𝑛 + 1)) ∈ ℕ0 → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
154	91, 153	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
155	152, 154	eqtrd 2772	. . . . . . . . . . . . . 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)))
156	104, 117, 155	3eqtr3d 2780	. . . . . . . . . . . . 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)))
157	105, 112	fsumcl 15668	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
158	105, 116	fsumcl 15668	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
159	119	recnd 11172	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
160	157, 158, 159	subaddd 11522	. . . . . . . . . . . . 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)))))
161	156, 160	mpbid 232	. . . . . . . . . . . 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))))
162	80, 161	eqeq12d 2753	. . . . . . . . . . 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)))))
163	62, 162	imbitrid 244	. . . . . . . . . 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)))))
164	163	ralimdva 3150	. . . . . . . . 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)))))
165	61, 164	syld 47	. . . . . . . 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)))))
166	165	ex 412	. . . . . . 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))))))
167	166	a2d 29	. . . . . 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))))))
168	8, 16, 24, 32, 53, 167	nn0ind 12599	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘)))))
169	168	imp 406	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
170		oveq2 7376	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
171	170	sumeq1d 15635	. . . . . 6 ⊢ (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
172	171	eqeq2d 2748	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
173	172	rspcv 3574	. . . 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)))))
174	169, 173	syl5 34	. . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
175	174	3impib 1117	. 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
176	175	3com12 1124	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3903  ifcif 4481   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Fincfn 8895  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   < clt 11178   ≤ cle 11179   − cmin 11376   / cdiv 11806  ℕcn 12157  2c2 12212  ℕ₀cn0 12413  ℤcz 12500  ℤ_≥cuz 12763  ...cfz 13435  ⌊cfl 13722  ↑cexp 13996  !cfa 14208  ♯chash 14265  Σcsu 15621   ∥ cdvds 16191  ℙcprime 16610   pCnt cpc 16776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-mod 13802  df-seq 13937  df-exp 13997  df-fac 14209  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622  df-dvds 16192  df-gcd 16434  df-prm 16611  df-pc 16777

This theorem is referenced by:  pcbc  16840