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Theorem pcfac 16223
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 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
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.
StepHypRef Expression
1 fveq2 6663 . . . . . . . 8 (𝑥 = 0 → (ℤ𝑥) = (ℤ‘0))
2 fveq2 6663 . . . . . . . . . 10 (𝑥 = 0 → (!‘𝑥) = (!‘0))
32oveq2d 7161 . . . . . . . . 9 (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4 fvoveq1 7168 . . . . . . . . . 10 (𝑥 = 0 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(0 / (𝑃𝑘))))
54sumeq2sdv 15049 . . . . . . . . 9 (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
63, 5eqeq12d 2834 . . . . . . . 8 (𝑥 = 0 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘)))))
71, 6raleqbidv 3399 . . . . . . 7 (𝑥 = 0 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘)))))
87imbi2d 342 . . . . . 6 (𝑥 = 0 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))))
9 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑛 → (ℤ𝑥) = (ℤ𝑛))
10 fveq2 6663 . . . . . . . . . 10 (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
1110oveq2d 7161 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12 fvoveq1 7168 . . . . . . . . . 10 (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑛 / (𝑃𝑘))))
1312sumeq2sdv 15049 . . . . . . . . 9 (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))))
1411, 13eqeq12d 2834 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
159, 14raleqbidv 3399 . . . . . . 7 (𝑥 = 𝑛 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
1615imbi2d 342 . . . . . 6 (𝑥 = 𝑛 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))))))
17 fveq2 6663 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (ℤ𝑥) = (ℤ‘(𝑛 + 1)))
18 fveq2 6663 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
1918oveq2d 7161 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20 fvoveq1 7168 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘((𝑛 + 1) / (𝑃𝑘))))
2120sumeq2sdv 15049 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))
2219, 21eqeq12d 2834 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
2317, 22raleqbidv 3399 . . . . . . 7 (𝑥 = (𝑛 + 1) → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
2423imbi2d 342 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))))
25 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑁 → (ℤ𝑥) = (ℤ𝑁))
26 fveq2 6663 . . . . . . . . . 10 (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
2726oveq2d 7161 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28 fvoveq1 7168 . . . . . . . . . 10 (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑁 / (𝑃𝑘))))
2928sumeq2sdv 15049 . . . . . . . . 9 (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
3027, 29eqeq12d 2834 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3125, 30raleqbidv 3399 . . . . . . 7 (𝑥 = 𝑁 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3231imbi2d 342 . . . . . 6 (𝑥 = 𝑁 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘)))) ↔ (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))))
33 fzfid 13329 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (1...𝑚) ∈ Fin)
34 sumz 15067 . . . . . . . . . 10 (((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
3534olcs 872 . . . . . . . . 9 ((1...𝑚) ∈ Fin → Σ𝑘 ∈ (1...𝑚)0 = 0)
3633, 35syl 17 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → Σ𝑘 ∈ (1...𝑚)0 = 0)
37 0nn0 11900 . . . . . . . . . 10 0 ∈ ℕ0
38 elfznn 12924 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
3938nnnn0d 11943 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
40 nn0uz 12268 . . . . . . . . . . . 12 0 = (ℤ‘0)
4139, 40eleqtrdi 2920 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘0))
4241adantl 482 . . . . . . . . . 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 16222 . . . . . . . . . 10 ((0 ∈ ℕ0𝑘 ∈ (ℤ‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃𝑘))) = 0)
4537, 42, 43, 44mp3an2i 1457 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃𝑘))) = 0)
4645sumeq2dv 15048 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
47 fac0 13624 . . . . . . . . . . 11 (!‘0) = 1
4847oveq2i 7156 . . . . . . . . . 10 (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
49 pc1 16180 . . . . . . . . . 10 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
5048, 49syl5eq 2865 . . . . . . . . 9 (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
5150adantr 481 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = 0)
5236, 46, 513eqtr4rd 2864 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
5352ralrimiva 3179 . . . . . 6 (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
54 nn0z 11993 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
5554adantr 481 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
56 uzid 12246 . . . . . . . . . . 11 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
57 peano2uz 12289 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑛) → (𝑛 + 1) ∈ (ℤ𝑛))
5855, 56, 573syl 18 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → (𝑛 + 1) ∈ (ℤ𝑛))
59 uzss 12253 . . . . . . . . . 10 ((𝑛 + 1) ∈ (ℤ𝑛) → (ℤ‘(𝑛 + 1)) ⊆ (ℤ𝑛))
60 ssralv 4030 . . . . . . . . . 10 ((ℤ‘(𝑛 + 1)) ⊆ (ℤ𝑛) → (∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) → ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
6158, 59, 603syl 18 . . . . . . . . 9 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → (∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) → ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
62 oveq1 7152 . . . . . . . . . . 11 ((𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) + (𝑃 pCnt (𝑛 + 1))))
63 simpll 763 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
64 facp1 13626 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
6563, 64syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
6665oveq2d 7161 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
67 simplr 765 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
68 faccl 13631 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
69 nnz 11992 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
70 nnne0 11659 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
7169, 70jca 512 . . . . . . . . . . . . . . 15 ((!‘𝑛) ∈ ℕ → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
7263, 68, 713syl 18 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
73 nn0p1nn 11924 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
74 nnz 11992 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
75 nnne0 11659 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
7674, 75jca 512 . . . . . . . . . . . . . . 15 ((𝑛 + 1) ∈ ℕ → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
7763, 73, 763syl 18 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
78 pcmul 16176 . . . . . . . . . . . . . 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))))
7967, 72, 77, 78syl3anc 1363 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))) = ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))))
8066, 79eqtr2d 2854 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
8163adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ0)
8281nn0zd 12073 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
83 prmnn 16006 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8483ad2antlr 723 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
85 nnexpcl 13430 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ)
8684, 39, 85syl2an 595 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃𝑘) ∈ ℕ)
87 fldivp1 16221 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℤ ∧ (𝑃𝑘) ∈ ℕ) → ((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = if((𝑃𝑘) ∥ (𝑛 + 1), 1, 0))
8882, 86, 87syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = if((𝑃𝑘) ∥ (𝑛 + 1), 1, 0))
89 elfzuz 12892 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘1))
9063, 73syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ)
9167, 90pccld 16175 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
9291nn0zd 12073 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
93 elfz5 12888 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ (ℤ‘1) ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
9489, 92, 93syl2anr 596 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
95 simpllr 772 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
9681, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℕ)
9796nnzd 12074 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
9839adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ0)
99 pcdvdsb 16193 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
10095, 97, 98, 99syl3anc 1363 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
10194, 100bitr2d 281 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑃𝑘) ∥ (𝑛 + 1) ↔ 𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1)))))
102101ifbid 4485 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → if((𝑃𝑘) ∥ (𝑛 + 1), 1, 0) = if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0))
10388, 102eqtrd 2853 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0))
104103sumeq2dv 15048 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0))
105 fzfid 13329 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
10663nn0red 11944 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
107 peano2re 10801 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
108106, 107syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℝ)
109108adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℝ)
110109, 86nndivred 11679 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃𝑘)) ∈ ℝ)
111110flcld 13156 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℤ)
112111zcnd 12076 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
113106adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℝ)
114113, 86nndivred 11679 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃𝑘)) ∈ ℝ)
115114flcld 13156 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℤ)
116115zcnd 12076 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
117105, 112, 116fsumsub 15131 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = (Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
118 fzfi 13328 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ Fin
11991nn0red 11944 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
120 eluzelz 12241 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
121120adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
122121zred 12075 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
123 prmuz2 16028 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
124123ad2antlr 723 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ (ℤ‘2))
12590nnnn0d 11943 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
126 bernneq3 13580 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ (ℤ‘2) ∧ (𝑛 + 1) ∈ ℕ0) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
127124, 125, 126syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
128119, 108letrid 10780 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
129128ord 858 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
13090nnzd 12074 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
131 pcdvdsb 16193 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ0) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13267, 130, 125, 131syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13384, 125nnexpcld 13594 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
134133nnzd 12074 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
135 dvdsle 15648 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
136134, 90, 135syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
137133nnred 11641 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
138137, 108lenltd 10774 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
139136, 138sylibd 240 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
140132, 139sylbid 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
141129, 140syld 47 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
142127, 141mt4d 117 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
143 eluzle 12244 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
144143adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
145119, 108, 122, 142, 144letrd 10785 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
146 eluz 12245 . . . . . . . . . . . . . . . . . . 19 (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
14792, 121, 146syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
148145, 147mpbird 258 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))))
149 fzss2 12935 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
150148, 149syl 17 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
151 sumhash 16220 . . . . . . . . . . . . . . . 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)))))
152118, 150, 151sylancr 587 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (♯‘(1...(𝑃 pCnt (𝑛 + 1)))))
153 hashfz1 13694 . . . . . . . . . . . . . . . 16 ((𝑃 pCnt (𝑛 + 1)) ∈ ℕ0 → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
15491, 153syl 17 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
155152, 154eqtrd 2853 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (𝑃 pCnt (𝑛 + 1)))
156104, 117, 1553eqtr3d 2861 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))) = (𝑃 pCnt (𝑛 + 1)))
157105, 112fsumcl 15078 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
158105, 116fsumcl 15078 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
159119recnd 10657 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
160157, 158, 159subaddd 11003 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))) = (𝑃 pCnt (𝑛 + 1)) ↔ (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) + (𝑃 pCnt (𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
161156, 160mpbid 233 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) + (𝑃 pCnt (𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))
16280, 161eqeq12d 2834 . . . . . . . . . . 11 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) + (𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
16362, 162syl5ib 245 . . . . . . . . . 10 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) → (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
164163ralimdva 3174 . . . . . . . . 9 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → (∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) → ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
16561, 164syld 47 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → (∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) → ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
166165ex 413 . . . . . . 7 (𝑛 ∈ ℕ0 → (𝑃 ∈ ℙ → (∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) → ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))))
167166a2d 29 . . . . . 6 (𝑛 ∈ ℕ0 → ((𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))) → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ‘(𝑛 + 1))(𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))))
1688, 16, 24, 32, 53, 167nn0ind 12065 . . . . 5 (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
169168imp 407 . . . 4 ((𝑁 ∈ ℕ0𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
170 oveq2 7153 . . . . . . 7 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
171170sumeq1d 15046 . . . . . 6 (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
172171eqeq2d 2829 . . . . 5 (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
173172rspcv 3615 . . . 4 (𝑀 ∈ (ℤ𝑁) → (∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
174169, 173syl5 34 . . 3 (𝑀 ∈ (ℤ𝑁) → ((𝑁 ∈ ℕ0𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
1751743impib 1108 . 2 ((𝑀 ∈ (ℤ𝑁) ∧ 𝑁 ∈ ℕ0𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
1761753com12 1115 1 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wss 3933  ifcif 4463   class class class wbr 5057  cfv 6348  (class class class)co 7145  Fincfn 8497  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  cz 11969  cuz 12231  ...cfz 12880  cfl 13148  cexp 13417  !cfa 13621  chash 13678  Σcsu 15030  cdvds 15595  cprime 16003   pCnt cpc 16161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-dvds 15596  df-gcd 15832  df-prm 16004  df-pc 16162
This theorem is referenced by:  pcbc  16224
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