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Theorem pcfac 16838

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 6892	. . . . . . . 8 ⊢ (𝑥 = 0 → (ℤ_≥‘𝑥) = (ℤ_≥‘0))
2		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
3	2	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4		fvoveq1 7436	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(0 / (𝑃↑𝑘))))
5	4	sumeq2sdv 15656	. . . . . . . . 9 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
6	3, 5	eqeq12d 2746	. . . . . . . 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 3340	. . . . . . 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 339	. . . . . 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 6892	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (ℤ_≥‘𝑥) = (ℤ_≥‘𝑛))
10		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
11	10	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12		fvoveq1 7436	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑛 / (𝑃↑𝑘))))
13	12	sumeq2sdv 15656	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))
14	11, 13	eqeq12d 2746	. . . . . . . 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 3340	. . . . . . 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 339	. . . . . 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 6892	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (ℤ_≥‘𝑥) = (ℤ_≥‘(𝑛 + 1)))
18		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
19	18	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20		fvoveq1 7436	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
21	20	sumeq2sdv 15656	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
22	19, 21	eqeq12d 2746	. . . . . . . 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 3340	. . . . . . 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 339	. . . . . 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 6892	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (ℤ_≥‘𝑥) = (ℤ_≥‘𝑁))
26		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
27	26	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28		fvoveq1 7436	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑁 / (𝑃↑𝑘))))
29	28	sumeq2sdv 15656	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
30	27, 29	eqeq12d 2746	. . . . . . . 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 3340	. . . . . . 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 339	. . . . . 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 13944	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → (1...𝑚) ∈ Fin)
34		sumz 15674	. . . . . . . . . 10 ⊢ (((1...𝑚) ⊆ (ℤ_≥‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
35	34	olcs 872	. . . . . . . . 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 12493	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
38		elfznn 13536	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
39	38	nnnn0d 12538	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ₀)
40		nn0uz 12870	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
41	39, 40	eleqtrdi 2841	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ_≥‘0))
42	41	adantl 480	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ (ℤ_≥‘0))
43		simpll 763	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
44		pcfaclem 16837	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
45	37, 42, 43, 44	mp3an2i 1464	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
46	45	sumeq2dv 15655	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
47		fac0 14242	. . . . . . . . . . 11 ⊢ (!‘0) = 1
48	47	oveq2i 7424	. . . . . . . . . 10 ⊢ (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
49		pc1 16794	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
50	48, 49	eqtrid 2782	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
51	50	adantr 479	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → (𝑃 pCnt (!‘0)) = 0)
52	36, 46, 51	3eqtr4rd 2781	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
53	52	ralrimiva 3144	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ_≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
54		nn0z 12589	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
55	54	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
56		uzid 12843	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
57		peano2uz 12891	. . . . . . . . . . 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 12851	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘(𝑛 + 1)) ⊆ (ℤ_≥‘𝑛))
60		ssralv 4051	. . . . . . . . . 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 7420	. . . . . . . . . . 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 763	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ₀)
64		facp1 14244	. . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
67		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
68		faccl 14249	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (!‘𝑛) ∈ ℕ)
69		nnz 12585	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
70		nnne0 12252	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
71	69, 70	jca 510	. . . . . . . . . . . . . . 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 12517	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
74		nnz 12585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
75		nnne0 12252	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
76	74, 75	jca 510	. . . . . . . . . . . . . . 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 16790	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
81	63	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ₀)
82	81	nn0zd 12590	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
83		prmnn 16617	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
84	83	ad2antlr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
85		nnexpcl 14046	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑃↑𝑘) ∈ ℕ)
86	84, 39, 85	syl2an 594	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃↑𝑘) ∈ ℕ)
87		fldivp1 16836	. . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . 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 13503	. . . . . . . . . . . . . . . . . . 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 16789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ₀)
92	91	nn0zd 12590	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
93		elfz5 13499	. . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 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 12591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
98	39	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ₀)
99		pcdvdsb 16808	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
100	95, 97, 98, 99	syl3anc 1369	. . . . . . . . . . . . . . . . . 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 4552	. . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . 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 15655	. . . . . . . . . . . . . 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 13944	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
106	63	nn0red 12539	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
107		peano2re 11393	. . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℝ)
110	109, 86	nndivred 12272	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℝ)
111	110	flcld 13769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℤ)
112	111	zcnd 12673	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
113	106	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℝ)
114	113, 86	nndivred 12272	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃↑𝑘)) ∈ ℝ)
115	114	flcld 13769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℤ)
116	115	zcnd 12673	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
117	105, 112, 116	fsumsub 15740	. . . . . . . . . . . . . 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 13943	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ∈ Fin
119	91	nn0red 12539	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
120		eluzelz 12838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
121	120	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
122	121	zred 12672	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
123		prmuz2 16639	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
124	123	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ (ℤ_≥‘2))
125	90	nnnn0d 12538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ₀)
126		bernneq3 14200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑛 + 1) ∈ ℕ₀) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
127	124, 125, 126	syl2anc 582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
128	119, 108	letrid 11372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
129	128	ord 860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
130	90	nnzd 12591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
131		pcdvdsb 16808	. . . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
133	84, 125	nnexpcld 14214	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
134	133	nnzd 12591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
135		dvdsle 16259	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
136	134, 90, 135	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
137	133	nnred 12233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
138	137, 108	lenltd 11366	. . . . . . . . . . . . . . . . . . . . . . 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 12841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
144	143	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
145	119, 108, 122, 142, 144	letrd 11377	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
146		eluz 12842	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ_≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
147	92, 121, 146	syl2anc 582	. . . . . . . . . . . . . . . . . 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 13547	. . . . . . . . . . . . . . . . 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 16835	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 14312	. . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 15685	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
158	105, 116	fsumcl 15685	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
159	119	recnd 11248	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
160	157, 158, 159	subaddd 11595	. . . . . . . . . . . . 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 2746	. . . . . . . . . . 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 3165	. . . . . . . . 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 411	. . . . . . 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 12663	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ_≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘)))))
169	168	imp 405	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ_≥‘𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
170		oveq2 7421	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
171	170	sumeq1d 15653	. . . . . 6 ⊢ (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
172	171	eqeq2d 2741	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
173	172	rspcv 3609	. . . 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 1114	. 2 ⊢ ((𝑀 ∈ (ℤ_≥‘𝑁) ∧ 𝑁 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
176	175	3com12 1121	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 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ⊆ wss 3949 ifcif 4529 class class class wbr 5149 ‘cfv 6544 (class class class)co 7413 Fincfn 8943 ℝcr 11113 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 < clt 11254 ≤ cle 11255 − cmin 11450 / cdiv 11877 ℕcn 12218 2c2 12273 ℕ₀cn0 12478 ℤcz 12564 ℤ_≥cuz 12828 ...cfz 13490 ⌊cfl 13761 ↑cexp 14033 !cfa 14239 ♯chash 14296 Σcsu 15638 ∥ cdvds 16203 ℙcprime 16614 pCnt cpc 16775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-z 12565 df-uz 12829 df-q 12939 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14034 df-fac 14240 df-hash 14297 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-dvds 16204 df-gcd 16442 df-prm 16615 df-pc 16776

This theorem is referenced by: pcbc 16839

W3C validator