Theorem pcfac 12257

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 5480	. . . . . . . 8 ⊢ (𝑥 = 0 → (ℤ≥‘𝑥) = (ℤ≥‘0))
2		fveq2 5480	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
3	2	oveq2d 5852	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4		fvoveq1 5859	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(0 / (𝑃↑𝑘))))
5	4	sumeq2sdv 11297	. . . . . . . . 9 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
6	3, 5	eqeq12d 2179	. . . . . . . 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 2671	. . . . . . 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 5480	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (ℤ≥‘𝑥) = (ℤ≥‘𝑛))
10		fveq2 5480	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
11	10	oveq2d 5852	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12		fvoveq1 5859	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑛 / (𝑃↑𝑘))))
13	12	sumeq2sdv 11297	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))
14	11, 13	eqeq12d 2179	. . . . . . . 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 2671	. . . . . . 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 5480	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (ℤ≥‘𝑥) = (ℤ≥‘(𝑛 + 1)))
18		fveq2 5480	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
19	18	oveq2d 5852	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20		fvoveq1 5859	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
21	20	sumeq2sdv 11297	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
22	19, 21	eqeq12d 2179	. . . . . . . 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 2671	. . . . . . 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 5480	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (ℤ≥‘𝑥) = (ℤ≥‘𝑁))
26		fveq2 5480	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
27	26	oveq2d 5852	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28		fvoveq1 5859	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑁 / (𝑃↑𝑘))))
29	28	sumeq2sdv 11297	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
30	27, 29	eqeq12d 2179	. . . . . . . 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 2671	. . . . . . 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 9209	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → 1 ∈ ℤ)
34		eluzelz 9466	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘0) → 𝑚 ∈ ℤ)
35	34	adantl 275	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → 𝑚 ∈ ℤ)
36	33, 35	fzfigd 10356	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (1...𝑚) ∈ Fin)
37		isumz 11316	. . . . . . . . . 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 726	. . . . . . . . 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 9120	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
41		elfznn 9979	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
42	41	nnnn0d 9158	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
43		nn0uz 9491	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
44	42, 43	eleqtrdi 2257	. . . . . . . . . . 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 519	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
47		pcfaclem 12256	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
48	40, 45, 46, 47	mp3an2i 1331	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
49	48	sumeq2dv 11295	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
50		fac0 10630	. . . . . . . . . . 11 ⊢ (!‘0) = 1
51	50	oveq2i 5847	. . . . . . . . . 10 ⊢ (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
52		pc1 12214	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
53	51, 52	syl5eq 2209	. . . . . . . . 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 2208	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
56	55	ralrimiva 2537	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
57		nn0z 9202	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
58	57	adantr 274	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
59		uzid 9471	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
60		peano2uz 9512	. . . . . . . . . . 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 9477	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘𝑛) → (ℤ≥‘(𝑛 + 1)) ⊆ (ℤ≥‘𝑛))
63		ssralv 3201	. . . . . . . . . 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 5843	. . . . . . . . . . 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 519	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
67		facp1 10632	. . . . . . . . . . . . . . 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 5852	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
70		simplr 520	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
71		faccl 10637	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
72		nnz 9201	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
73		nnne0 8876	. . . . . . . . . . . . . . . 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 9144	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
77		nnz 9201	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
78		nnne0 8876	. . . . . . . . . . . . . . . 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 12210	. . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . 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 2198	. . . . . . . . . . . 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 9302	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
86		prmnn 12021	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
87	86	ad2antlr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
88		nnexpcl 10458	. . . . . . . . . . . . . . . . . 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 12255	. . . . . . . . . . . . . . . . 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 9947	. . . . . . . . . . . . . . . . . . 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 12209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
95	94	nn0zd 9302	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
96		elfz5 9943	. . . . . . . . . . . . . . . . . . 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 524	. . . . . . . . . . . . . . . . . . 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 9303	. . . . . . . . . . . . . . . . . . 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 12228	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
103	98, 100, 101, 102	syl3anc 1227	. . . . . . . . . . . . . . . . . 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 3536	. . . . . . . . . . . . . . . 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 2197	. . . . . . . . . . . . . . 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 11295	. . . . . . . . . . . . . 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 9209	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 1 ∈ ℤ)
109		eluzelz 9466	. . . . . . . . . . . . . . . . 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 10356	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
112		znq 9553	. . . . . . . . . . . . . . . . . 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 10202	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℤ)
115	114	zcnd 9305	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
116		znq 9553	. . . . . . . . . . . . . . . . . 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 10202	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℤ)
119	118	zcnd 9305	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
120	111, 115, 119	fsumsub 11379	. . . . . . . . . . . . . 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 9159	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
122	66	nn0red 9159	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
123		peano2re 8025	. . . . . . . . . . . . . . . . . . . 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 9304	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
126	93	nnzd 9303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
127		zdcle 9258	. . . . . . . . . . . . . . . . . . . . 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 9226	. . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
132	93	nnnn0d 9158	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
133		pcdvdsb 12228	. . . . . . . . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
135	87, 132	nnexpcld 10599	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
136	135	nnzd 9303	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
137		dvdsle 11767	. . . . . . . . . . . . . . . . . . . . . . . 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 8861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
140	139, 124	lenltd 8007	. . . . . . . . . . . . . . . . . . . . . . 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 12042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
145	144	ad2antlr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑃 ∈ (ℤ≥‘2))
146		bernneq3 10566	. . . . . . . . . . . . . . . . . . . . 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 843	. . . . . . . . . . . . . . . . . . . 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 9469	. . . . . . . . . . . . . . . . . . . 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 8013	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
153		eluz 9470	. . . . . . . . . . . . . . . . . . 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 9989	. . . . . . . . . . . . . . . . 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 9951	. . . . . . . . . . . . . . . . . . 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 9209	. . . . . . . . . . . . . . . . . 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 9965	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
163	159, 160, 161, 162	syl3anc 1227	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
164	163	ralrimiva 2537	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → ∀𝑗 ∈ (1...𝑚)DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
165		sumhashdc 12254	. . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . 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 10685	. . . . . . . . . . . . . . . 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 2197	. . . . . . . . . . . . . 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 2205	. . . . . . . . . . . . 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 11327	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
172	111, 119	fsumcl 11327	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
173	94	nn0cnd 9160	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
174	171, 172, 173	subaddd 8218	. . . . . . . . . . . . 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 2179	. . . . . . . . . . 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 2531	. . . . . . . . 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 9296	. . . . 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 5844	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
185	184	sumeq1d 11293	. . . . . 6 ⊢ (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
186	185	eqeq2d 2176	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
187	186	rspcv 2821	. . . 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 1190	. 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
190	189	3com12 1196	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 698  DECID wdc 824   ∧ w3a 967   = wceq 1342   ∈ wcel 2135   ≠ wne 2334  ∀wral 2442   ⊆ wss 3111  ifcif 3515   class class class wbr 3976  ‘cfv 5182  (class class class)co 5836  Fincfn 6697  ℝcr 7743  0cc0 7744  1c1 7745   + caddc 7747   · cmul 7749   < clt 7924   ≤ cle 7925   − cmin 8060   / cdiv 8559  ℕcn 8848  2c2 8899  ℕ₀cn0 9105  ℤcz 9182  ℤ_≥cuz 9457  ℚcq 9548  ...cfz 9935  ⌊cfl 10193  ↑cexp 10444  !cfa 10627  ♯chash 10677  Σcsu 11280   ∥ cdvds 11713  ℙcprime 12018   pCnt cpc 12193

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-isom 5191  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-frec 6350  df-1o 6375  df-2o 6376  df-oadd 6379  df-er 6492  df-en 6698  df-dom 6699  df-fin 6700  df-sup 6940  df-inf 6941  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-q 9549  df-rp 9581  df-fz 9936  df-fzo 10068  df-fl 10195  df-mod 10248  df-seqfrec 10371  df-exp 10445  df-fac 10628  df-ihash 10678  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-clim 11206  df-sumdc 11281  df-dvds 11714  df-gcd 11861  df-prm 12019  df-pc 12194

This theorem is referenced by:  pcbc  12258