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Theorem pcfac 12351

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 5517	. . . . . . . 8 ⊢ (𝑥 = 0 → (ℤ_≥‘𝑥) = (ℤ_≥‘0))
2		fveq2 5517	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (!‘𝑥) = (!‘0))
3	2	oveq2d 5894	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4		fvoveq1 5901	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(0 / (𝑃↑𝑘))))
5	4	sumeq2sdv 11381	. . . . . . . . 9 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
6	3, 5	eqeq12d 2192	. . . . . . . 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 2685	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (ℤ_≥‘𝑥) = (ℤ_≥‘𝑛))
10		fveq2 5517	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
11	10	oveq2d 5894	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12		fvoveq1 5901	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑛 / (𝑃↑𝑘))))
13	12	sumeq2sdv 11381	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))))
14	11, 13	eqeq12d 2192	. . . . . . . 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 2685	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (ℤ_≥‘𝑥) = (ℤ_≥‘(𝑛 + 1)))
18		fveq2 5517	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
19	18	oveq2d 5894	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20		fvoveq1 5901	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
21	20	sumeq2sdv 11381	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))))
22	19, 21	eqeq12d 2192	. . . . . . . 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 2685	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (ℤ_≥‘𝑥) = (ℤ_≥‘𝑁))
26		fveq2 5517	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
27	26	oveq2d 5894	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28		fvoveq1 5901	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃↑𝑘))) = (⌊‘(𝑁 / (𝑃↑𝑘))))
29	28	sumeq2sdv 11381	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))))
30	27, 29	eqeq12d 2192	. . . . . . . 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 2685	. . . . . . 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 9283	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → 1 ∈ ℤ)
34		eluzelz 9540	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘0) → 𝑚 ∈ ℤ)
35	34	adantl 277	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → 𝑚 ∈ ℤ)
36	33, 35	fzfigd 10434	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → (1...𝑚) ∈ Fin)
37		isumz 11400	. . . . . . . . . 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 736	. . . . . . . . 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 9194	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
41		elfznn 10057	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
42	41	nnnn0d 9232	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ₀)
43		nn0uz 9565	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
44	42, 43	eleqtrdi 2270	. . . . . . . . . . 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 527	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
47		pcfaclem 12350	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
48	40, 45, 46, 47	mp3an2i 1342	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃↑𝑘))) = 0)
49	48	sumeq2dv 11379	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
50		fac0 10711	. . . . . . . . . . 11 ⊢ (!‘0) = 1
51	50	oveq2i 5889	. . . . . . . . . 10 ⊢ (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
52		pc1 12308	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
53	51, 52	eqtrid 2222	. . . . . . . . 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 2221	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ_≥‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
56	55	ralrimiva 2550	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ_≥‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃↑𝑘))))
57		nn0z 9276	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
58	57	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
59		uzid 9545	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
60		peano2uz 9586	. . . . . . . . . . 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 9551	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘(𝑛 + 1)) ⊆ (ℤ_≥‘𝑛))
63		ssralv 3221	. . . . . . . . . 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 5885	. . . . . . . . . . 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 527	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ₀)
67		facp1 10713	. . . . . . . . . . . . . . 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 5894	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
70		simplr 528	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
71		faccl 10718	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (!‘𝑛) ∈ ℕ)
72		nnz 9275	. . . . . . . . . . . . . . . 16 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
73		nnne0 8950	. . . . . . . . . . . . . . . 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 9218	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
77		nnz 9275	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
78		nnne0 8950	. . . . . . . . . . . . . . . 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 12304	. . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . 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 2211	. . . . . . . . . . . 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 9376	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
86		prmnn 12113	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
87	86	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
88		nnexpcl 10536	. . . . . . . . . . . . . . . . . 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 12349	. . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . 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 10024	. . . . . . . . . . . . . . . . . . 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 12303	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ₀)
95	94	nn0zd 9376	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
96		elfz5 10020	. . . . . . . . . . . . . . . . . . 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 534	. . . . . . . . . . . . . . . . . . 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 9377	. . . . . . . . . . . . . . . . . . 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 12322	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑘) ∥ (𝑛 + 1)))
103	98, 100, 101, 102	syl3anc 1238	. . . . . . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . . . . 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 2210	. . . . . . . . . . . . . . 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 11379	. . . . . . . . . . . . . 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 9283	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 1 ∈ ℤ)
109		eluzelz 9540	. . . . . . . . . . . . . . . . 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 10434	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
112		znq 9627	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 + 1) ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℚ)
113	100, 89, 112	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃↑𝑘)) ∈ ℚ)
114	113	flqcld 10280	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℤ)
115	114	zcnd 9379	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
116		znq 9627	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℤ ∧ (𝑃↑𝑘) ∈ ℕ) → (𝑛 / (𝑃↑𝑘)) ∈ ℚ)
117	85, 89, 116	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃↑𝑘)) ∈ ℚ)
118	117	flqcld 10280	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℤ)
119	118	zcnd 9379	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
120	111, 115, 119	fsumsub 11463	. . . . . . . . . . . . . 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 9233	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
122	66	nn0red 9233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
123		peano2re 8096	. . . . . . . . . . . . . . . . . . . 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 9378	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
126	93	nnzd 9377	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
127		zdcle 9332	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℤ) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
128	95, 126, 127	syl2anc 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
129		zletric 9300	. . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
131	130	ord 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
132	93	nnnn0d 9232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ₀)
133		pcdvdsb 12322	. . . . . . . . . . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
135	87, 132	nnexpcld 10679	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
136	135	nnzd 9377	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
137		dvdsle 11853	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
138	136, 93, 137	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
139	135	nnred 8935	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
140	139, 124	lenltd 8078	. . . . . . . . . . . . . . . . . . . . . . 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 12134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
145	144	ad2antlr 489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑃 ∈ (ℤ_≥‘2))
146		bernneq3 10646	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑛 + 1) ∈ ℕ₀) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
147	145, 132, 146	syl2anc 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
148		condc 853	. . . . . . . . . . . . . . . . . . . 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 9543	. . . . . . . . . . . . . . . . . . . 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 8084	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
153		eluz 9544	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ_≥‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
154	95, 110, 153	syl2anc 411	. . . . . . . . . . . . . . . . . 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 10067	. . . . . . . . . . . . . . . . 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 10028	. . . . . . . . . . . . . . . . . . 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 9283	. . . . . . . . . . . . . . . . . 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 10043	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
163	159, 160, 161, 162	syl3anc 1238	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
164	163	ralrimiva 2550	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → ∀𝑗 ∈ (1...𝑚)DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
165		sumhashdc 12348	. . . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . 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 10766	. . . . . . . . . . . . . . . 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 2210	. . . . . . . . . . . . . 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 2218	. . . . . . . . . . . . 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 11411	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃↑𝑘))) ∈ ℂ)
172	111, 119	fsumcl 11411	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃↑𝑘))) ∈ ℂ)
173	94	nn0cnd 9234	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
174	171, 172, 173	subaddd 8289	. . . . . . . . . . . . 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 2192	. . . . . . . . . . 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 2544	. . . . . . . . 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 9370	. . . . 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 5886	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
185	184	sumeq1d 11377	. . . . . 6 ⊢ (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
186	185	eqeq2d 2189	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘)))))
187	186	rspcv 2839	. . . 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 1201	. 2 ⊢ ((𝑀 ∈ (ℤ_≥‘𝑁) ∧ 𝑁 ∈ ℕ₀ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))
190	189	3com12 1207	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ (ℤ_≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃↑𝑘))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ⊆ wss 3131 ifcif 3536 class class class wbr 4005 ‘cfv 5218 (class class class)co 5878 Fincfn 6743 ℝcr 7813 0cc0 7814 1c1 7815 + caddc 7817 · cmul 7819 < clt 7995 ≤ cle 7996 − cmin 8131 / cdiv 8632 ℕcn 8922 2c2 8973 ℕ₀cn0 9179 ℤcz 9256 ℤ_≥cuz 9531 ℚcq 9622 ...cfz 10011 ⌊cfl 10271 ↑cexp 10522 !cfa 10708 ♯chash 10758 Σcsu 11364 ∥ cdvds 11797 ℙcprime 12110 pCnt cpc 12287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-irdg 6374 df-frec 6395 df-1o 6420 df-2o 6421 df-oadd 6424 df-er 6538 df-en 6744 df-dom 6745 df-fin 6746 df-sup 6986 df-inf 6987 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-q 9623 df-rp 9657 df-fz 10012 df-fzo 10146 df-fl 10273 df-mod 10326 df-seqfrec 10449 df-exp 10523 df-fac 10709 df-ihash 10759 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-clim 11290 df-sumdc 11365 df-dvds 11798 df-gcd 11947 df-prm 12111 df-pc 12288

This theorem is referenced by: pcbc 12352

W3C validator