Theorem pcfac 13152

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 5695	. . . . . . . 8 ⊢ (𝑥 = 0 → (ℤ≥‘𝑥) = (ℤ≥‘0))
2		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
3	2	oveq2d 6101	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4		fvoveq1 6108	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(0 / (𝑃↑𝑘))))
5	4	sumeq2sdv 12155	. . . . . . . . 9 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
6	3, 5	eqeq12d 2253	. . . . . . . 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 2765	. . . . . . 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 230	. . . . . 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 5695	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (ℤ≥‘𝑥) = (ℤ≥‘𝑛))
10		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
11	10	oveq2d 6101	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12		fvoveq1 6108	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑛 / (𝑃↑𝑘))))
13	12	sumeq2sdv 12155	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))
14	11, 13	eqeq12d 2253	. . . . . . . 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 2765	. . . . . . 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 230	. . . . . 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 5695	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (ℤ≥‘𝑥) = (ℤ≥‘(𝑛 + 1)))
18		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
19	18	oveq2d 6101	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20		fvoveq1 6108	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
21	20	sumeq2sdv 12155	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
22	19, 21	eqeq12d 2253	. . . . . . . 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 2765	. . . . . . 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 230	. . . . . 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 5695	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (ℤ≥‘𝑥) = (ℤ≥‘𝑁))
26		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
27	26	oveq2d 6101	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28		fvoveq1 6108	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑁 / (𝑃↑𝑘))))
29	28	sumeq2sdv 12155	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
30	27, 29	eqeq12d 2253	. . . . . . . 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 2765	. . . . . . 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 230	. . . . . 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 9676	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → 1 ∈ ℤ)
34		eluzelz 9941	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘0) → 𝑚 ∈ ℤ)
35	34	adantl 277	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → 𝑚 ∈ ℤ)
36	33, 35	fzfigd 10883	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (1...𝑚) ∈ Fin)
37		isumz 12175	. . . . . . . . . 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 748	. . . . . . . . 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 9583	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
41		elfznn 10471	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
42	41	nnnn0d 9625	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
43		nn0uz 9967	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
44	42, 43	eleqtrdi 2331	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ≥‘0))
45	44	adantl 277	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ (ℤ≥‘0))
46		simpll 531	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
47		pcfaclem 13151	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
48	40, 45, 46, 47	mp3an2i 1383	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
49	48	sumeq2dv 12153	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
50		fac0 11182	. . . . . . . . . . 11 ⊢ (!‘0) = 1
51	50	oveq2i 6096	. . . . . . . . . 10 ⊢ (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
52		pc1 13107	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
53	51, 52	eqtrid 2283	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
54	53	adantr 276	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝑃 pCnt (!‘0)) = 0)
55	39, 49, 54	3eqtr4rd 2282	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
56	55	ralrimiva 2623	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
57		nn0z 9669	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
58	57	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
59		uzid 9946	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
60		peano2uz 9993	. . . . . . . . . . 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 9953	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘𝑛) → (ℤ≥‘(𝑛 + 1)) ⊆ (ℤ≥‘𝑛))
63		ssralv 3312	. . . . . . . . . 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 6092	. . . . . . . . . . 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 531	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
67		facp1 11184	. . . . . . . . . . . . . . 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 6101	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
70		simplr 533	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
71		faccl 11189	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
72		nnz 9668	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
73		nnne0 9335	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
74	72, 73	jca 306	. . . . . . . . . . . . . . 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 9607	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
77		nnz 9668	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
78		nnne0 9335	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
79	77, 78	jca 306	. . . . . . . . . . . . . . 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 13103	. . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . 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 2272	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
84	66	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ0)
85	84	nn0zd 9771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
86		prmnn 12907	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
87	86	ad2antlr 493	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
88		nnexpcl 11004	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃↑𝑘) ∈ ℕ)
89	87, 42, 88	syl2an 289	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃↑𝑘) ∈ ℕ)
90		fldivp1 13150	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 10435	. . . . . . . . . . . . . . . . . . 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 13102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
95	94	nn0zd 9771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
96		elfz5 10431	. . . . . . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . . . . . 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 540	. . . . . . . . . . . . . . . . . . 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 9772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
101	42	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ0)
102		pcdvdsb 13122	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
103	98, 100, 101, 102	syl3anc 1278	. . . . . . . . . . . . . . . . . 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 189	. . . . . . . . . . . . . . . . 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 3662	. . . . . . . . . . . . . . . 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 2271	. . . . . . . . . . . . . . 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 12153	. . . . . . . . . . . . . 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 9676	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 1 ∈ ℤ)
109		eluzelz 9941	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
110	109	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
111	108, 110	fzfigd 10883	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
112		znq 10034	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 + 1) ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℚ)
113	100, 89, 112	syl2anc 415	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℚ)
114	113	flqcld 10725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℤ)
115	114	zcnd 9774	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
116		znq 10034	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → (𝑛 / (𝑃↑𝑘)) ∈ ℚ)
117	85, 89, 116	syl2anc 415	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃↑𝑘)) ∈ ℚ)
118	117	flqcld 10725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℤ)
119	118	zcnd 9774	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
120	111, 115, 119	fsumsub 12238	. . . . . . . . . . . . . 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 9626	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
122	66	nn0red 9626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
123		peano2re 8464	. . . . . . . . . . . . . . . . . . . 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 9773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
126	93	nnzd 9772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
127		zdcle 9726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℤ) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
128	95, 126, 127	syl2anc 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
129		zletric 9693	. . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
131	130	ord 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
132	93	nnnn0d 9625	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
133		pcdvdsb 13122	. . . . . . . . . . . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
135	87, 132	nnexpcld 11148	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
136	135	nnzd 9772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
137		dvdsle 12630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
138	136, 93, 137	syl2anc 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
139	135	nnred 9320	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
140	139, 124	lenltd 8446	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
141	138, 140	sylibd 149	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
142	134, 141	sylbid 150	. . . . . . . . . . . . . . . . . . . . 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 12929	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
145	144	ad2antlr 493	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ (ℤ≥‘2))
146		bernneq3 11115	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑛 + 1) ∈ ℕ0) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
147	145, 132, 146	syl2anc 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
148		condc 865	. . . . . . . . . . . . . . . . . . . 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 9944	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
151	150	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
152	121, 124, 125, 149, 151	letrd 8452	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
153		eluz 9945	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
154	95, 110, 153	syl2anc 415	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
155	152, 154	mpbird 167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ (ℤ≥‘(𝑃 pCnt (𝑛 + 1))))
156		fzss2 10481	. . . . . . . . . . . . . . . . 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 10439	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℤ)
159	158	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℤ)
160		1zzd 9676	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
161	95	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
162		fzdcel 10455	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
163	159, 160, 161, 162	syl3anc 1278	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
164	163	ralrimiva 2623	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ∀𝑗 ∈ (1...𝑚)DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
165		sumhashdc 13149	. . . . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . . 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 11238	. . . . . . . . . . . . . . . 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 2271	. . . . . . . . . . . . . 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 2279	. . . . . . . . . . . . 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 12186	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
172	111, 119	fsumcl 12186	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
173	94	nn0cnd 9627	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
174	171, 172, 173	subaddd 8657	. . . . . . . . . . . . 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 147	. . . . . . . . . . . 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 2253	. . . . . . . . . . 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	imbitrid 154	. . . . . . . . . 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 2617	. . . . . . . . 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 115	. . . . . . 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 9765	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘)))))
183	182	imp 124	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
184		oveq2 6093	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
185	184	sumeq1d 12151	. . . . . 6 ⊢ (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
186	185	eqeq2d 2250	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
187	186	rspcv 2925	. . . 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 1232	. 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
190	189	3com12 1238	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528   ⊆ wss 3220  ifcif 3638   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  Fincfn 7022  ℝcr 8179  0cc0 8180  1c1 8181   + caddc 8183   · cmul 8185   < clt 8361   ≤ cle 8362   − cmin 8499   / cdiv 9005  ℕcn 9307  2c2 9358  ℕ₀cn0 9568  ℤcz 9649  ℤ_≥cuz 9931  ℚcq 10029  ...cfz 10422  ⌊cfl 10714  ↑cexp 10990  !cfa 11179  ♯chash 11230  Σcsu 12138   ∥ cdvds 12573  ℙcprime 12904   pCnt cpc 13086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7325  df-inf 7326  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-fz 10423  df-fzo 10561  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-ihash 11231  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139  df-dvds 12574  df-gcd 12750  df-prm 12905  df-pc 13087

This theorem is used by:  pcbc  13153