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Theorem pcfac 16828

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 6888	. . . . . . . 8 ⊢ (𝑥 = 0 → (ℤ_≥‘𝑥) = (ℤ_≥‘0))
2		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
3	2	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4		fvoveq1 7428	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(0 / (𝑃↑𝑘))))
5	4	sumeq2sdv 15646	. . . . . . . . 9 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
6	3, 5	eqeq12d 2748	. . . . . . . 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 3342	. . . . . . 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 6888	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (ℤ_≥‘𝑥) = (ℤ_≥‘𝑛))
10		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
11	10	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12		fvoveq1 7428	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑛 / (𝑃↑𝑘))))
13	12	sumeq2sdv 15646	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))
14	11, 13	eqeq12d 2748	. . . . . . . 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 3342	. . . . . . 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 6888	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (ℤ_≥‘𝑥) = (ℤ_≥‘(𝑛 + 1)))
18		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
19	18	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20		fvoveq1 7428	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
21	20	sumeq2sdv 15646	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
22	19, 21	eqeq12d 2748	. . . . . . . 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 3342	. . . . . . 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 6888	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (ℤ_≥‘𝑥) = (ℤ_≥‘𝑁))
26		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
27	26	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28		fvoveq1 7428	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑁 / (𝑃↑𝑘))))
29	28	sumeq2sdv 15646	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
30	27, 29	eqeq12d 2748	. . . . . . . 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 3342	. . . . . . 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 13934	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → (1...𝑚) ∈ Fin)
34		sumz 15664	. . . . . . . . . 10 ⊢ (((1...𝑚) ⊆ (ℤ_≥‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
35	34	olcs 874	. . . . . . . . 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 12483	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
38		elfznn 13526	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
39	38	nnnn0d 12528	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ₀)
40		nn0uz 12860	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
41	39, 40	eleqtrdi 2843	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ_≥‘0))
42	41	adantl 482	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ (ℤ_≥‘0))
43		simpll 765	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
44		pcfaclem 16827	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
45	37, 42, 43, 44	mp3an2i 1466	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
46	45	sumeq2dv 15645	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
47		fac0 14232	. . . . . . . . . . 11 ⊢ (!‘0) = 1
48	47	oveq2i 7416	. . . . . . . . . 10 ⊢ (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
49		pc1 16784	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
50	48, 49	eqtrid 2784	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
51	50	adantr 481	. . . . . . . 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 3146	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ_≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
54		nn0z 12579	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
55	54	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
56		uzid 12833	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
57		peano2uz 12881	. . . . . . . . . . 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 12841	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘(𝑛 + 1)) ⊆ (ℤ_≥‘𝑛))
60		ssralv 4049	. . . . . . . . . 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 7412	. . . . . . . . . . 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 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ₀)
64		facp1 14234	. . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
67		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
68		faccl 14239	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (!‘𝑛) ∈ ℕ)
69		nnz 12575	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
70		nnne0 12242	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
71	69, 70	jca 512	. . . . . . . . . . . . . . 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 12507	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
74		nnz 12575	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
75		nnne0 12242	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
76	74, 75	jca 512	. . . . . . . . . . . . . . 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 16780	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ₀)
82	81	nn0zd 12580	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
83		prmnn 16607	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
84	83	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
85		nnexpcl 14036	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑃↑𝑘) ∈ ℕ)
86	84, 39, 85	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃↑𝑘) ∈ ℕ)
87		fldivp1 16826	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 13493	. . . . . . . . . . . . . . . . . . 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 16779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ₀)
92	91	nn0zd 12580	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
93		elfz5 13489	. . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . 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 12581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
98	39	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ₀)
99		pcdvdsb 16798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
100	95, 97, 98, 99	syl3anc 1371	. . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . 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 4550	. . . . . . . . . . . . . . . 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 15645	. . . . . . . . . . . . . 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 13934	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
106	63	nn0red 12529	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
107		peano2re 11383	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℝ)
110	109, 86	nndivred 12262	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℝ)
111	110	flcld 13759	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℤ)
112	111	zcnd 12663	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
113	106	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℝ)
114	113, 86	nndivred 12262	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃↑𝑘)) ∈ ℝ)
115	114	flcld 13759	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℤ)
116	115	zcnd 12663	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
117	105, 112, 116	fsumsub 15730	. . . . . . . . . . . . . 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 13933	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ∈ Fin
119	91	nn0red 12529	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
120		eluzelz 12828	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
121	120	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
122	121	zred 12662	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
123		prmuz2 16629	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
124	123	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ (ℤ_≥‘2))
125	90	nnnn0d 12528	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ₀)
126		bernneq3 14190	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑛 + 1) ∈ ℕ₀) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
127	124, 125, 126	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
128	119, 108	letrid 11362	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
129	128	ord 862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
130	90	nnzd 12581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
131		pcdvdsb 16798	. . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
133	84, 125	nnexpcld 14204	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
134	133	nnzd 12581	. . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
137	133	nnred 12223	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
138	137, 108	lenltd 11356	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
139	136, 138	sylibd 238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
140	132, 139	sylbid 239	. . . . . . . . . . . . . . . . . . . . 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 12831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
144	143	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
145	119, 108, 122, 142, 144	letrd 11367	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
146		eluz 12832	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ_≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
147	92, 121, 146	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑚 ∈ (ℤ_≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
148	145, 147	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑚 ∈ (ℤ_≥‘(𝑃 pCnt (𝑛 + 1))))
149		fzss2 13537	. . . . . . . . . . . . . . . . 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 16825	. . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . 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 14302	. . . . . . . . . . . . . . . 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 15675	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
158	105, 116	fsumcl 15675	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
159	119	recnd 11238	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
160	157, 158, 159	subaddd 11585	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 2748	. . . . . . . . . . 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 243	. . . . . . . . . 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 3167	. . . . . . . . 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 413	. . . . . . 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 12653	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ_≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘)))))
169	168	imp 407	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ_≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
170		oveq2 7413	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
171	170	sumeq1d 15643	. . . . . 6 ⊢ (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
172	171	eqeq2d 2743	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
173	172	rspcv 3608	. . . 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 1116	. 2 ⊢ ((𝑀 ∈ (ℤ_≥‘𝑁) ∧ 𝑁 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
176	175	3com12 1123	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ (ℤ_≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ⌊cfl 13751 ↑cexp 14023 !cfa 14229 ♯chash 14286 Σcsu 15628 ∥ cdvds 16193 ℙcprime 16604 pCnt cpc 16765

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766

This theorem is referenced by: pcbc 16829

W3C validator