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Theorem pcfac 16938
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 6905 . . . . . . . 8 (𝑥 = 0 → (ℤ𝑥) = (ℤ‘0))
2 fveq2 6905 . . . . . . . . . 10 (𝑥 = 0 → (!‘𝑥) = (!‘0))
32oveq2d 7448 . . . . . . . . 9 (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4 fvoveq1 7455 . . . . . . . . . 10 (𝑥 = 0 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(0 / (𝑃𝑘))))
54sumeq2sdv 15740 . . . . . . . . 9 (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
63, 5eqeq12d 2752 . . . . . . . 8 (𝑥 = 0 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘)))))
71, 6raleqbidv 3345 . . . . . . 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 340 . . . . . 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 6905 . . . . . . . 8 (𝑥 = 𝑛 → (ℤ𝑥) = (ℤ𝑛))
10 fveq2 6905 . . . . . . . . . 10 (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
1110oveq2d 7448 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12 fvoveq1 7455 . . . . . . . . . 10 (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑛 / (𝑃𝑘))))
1312sumeq2sdv 15740 . . . . . . . . 9 (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))))
1411, 13eqeq12d 2752 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
159, 14raleqbidv 3345 . . . . . . 7 (𝑥 = 𝑛 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
1615imbi2d 340 . . . . . 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 6905 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (ℤ𝑥) = (ℤ‘(𝑛 + 1)))
18 fveq2 6905 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
1918oveq2d 7448 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20 fvoveq1 7455 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘((𝑛 + 1) / (𝑃𝑘))))
2120sumeq2sdv 15740 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))
2219, 21eqeq12d 2752 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
2317, 22raleqbidv 3345 . . . . . . 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 340 . . . . . 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 6905 . . . . . . . 8 (𝑥 = 𝑁 → (ℤ𝑥) = (ℤ𝑁))
26 fveq2 6905 . . . . . . . . . 10 (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
2726oveq2d 7448 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28 fvoveq1 7455 . . . . . . . . . 10 (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑁 / (𝑃𝑘))))
2928sumeq2sdv 15740 . . . . . . . . 9 (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
3027, 29eqeq12d 2752 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3125, 30raleqbidv 3345 . . . . . . 7 (𝑥 = 𝑁 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3231imbi2d 340 . . . . . 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 14015 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (1...𝑚) ∈ Fin)
34 sumz 15759 . . . . . . . . . 10 (((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
3534olcs 876 . . . . . . . . 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 12543 . . . . . . . . . 10 0 ∈ ℕ0
38 elfznn 13594 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
3938nnnn0d 12589 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
40 nn0uz 12921 . . . . . . . . . . . 12 0 = (ℤ‘0)
4139, 40eleqtrdi 2850 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘0))
4241adantl 481 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ (ℤ‘0))
43 simpll 766 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
44 pcfaclem 16937 . . . . . . . . . 10 ((0 ∈ ℕ0𝑘 ∈ (ℤ‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃𝑘))) = 0)
4537, 42, 43, 44mp3an2i 1467 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃𝑘))) = 0)
4645sumeq2dv 15739 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
47 fac0 14316 . . . . . . . . . . 11 (!‘0) = 1
4847oveq2i 7443 . . . . . . . . . 10 (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
49 pc1 16894 . . . . . . . . . 10 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
5048, 49eqtrid 2788 . . . . . . . . 9 (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
5150adantr 480 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = 0)
5236, 46, 513eqtr4rd 2787 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
5352ralrimiva 3145 . . . . . 6 (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
54 nn0z 12640 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
5554adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
56 uzid 12894 . . . . . . . . . . 11 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
57 peano2uz 12944 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑛) → (𝑛 + 1) ∈ (ℤ𝑛))
5855, 56, 573syl 18 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → (𝑛 + 1) ∈ (ℤ𝑛))
59 uzss 12902 . . . . . . . . . 10 ((𝑛 + 1) ∈ (ℤ𝑛) → (ℤ‘(𝑛 + 1)) ⊆ (ℤ𝑛))
60 ssralv 4051 . . . . . . . . . 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 7439 . . . . . . . . . . 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 766 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
64 facp1 14318 . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
67 simplr 768 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
68 faccl 14323 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
69 nnz 12636 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
70 nnne0 12301 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
7169, 70jca 511 . . . . . . . . . . . . . . 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 12567 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
74 nnz 12636 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
75 nnne0 12301 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
7674, 75jca 511 . . . . . . . . . . . . . . 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 16890 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))) = ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))))
8066, 79eqtr2d 2777 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
8163adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ0)
8281nn0zd 12641 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
83 prmnn 16712 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8483ad2antlr 727 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
85 nnexpcl 14116 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ)
8684, 39, 85syl2an 596 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃𝑘) ∈ ℕ)
87 fldivp1 16936 . . . . . . . . . . . . . . . . 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 13561 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘1))
9063, 73syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ)
9167, 90pccld 16889 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
9291nn0zd 12641 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
93 elfz5 13557 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ (ℤ‘1) ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
9489, 92, 93syl2anr 597 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
95 simpllr 775 . . . . . . . . . . . . . . . . . . 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 12642 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
9839adantl 481 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ0)
99 pcdvdsb 16908 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
10095, 97, 98, 99syl3anc 1372 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
10194, 100bitr2d 280 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑃𝑘) ∥ (𝑛 + 1) ↔ 𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1)))))
102101ifbid 4548 . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . 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 15739 . . . . . . . . . . . . . 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 14015 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
10663nn0red 12590 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
107 peano2re 11435 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
108106, 107syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℝ)
109108adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℝ)
110109, 86nndivred 12321 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃𝑘)) ∈ ℝ)
111110flcld 13839 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℤ)
112111zcnd 12725 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
113106adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℝ)
114113, 86nndivred 12321 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃𝑘)) ∈ ℝ)
115114flcld 13839 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℤ)
116115zcnd 12725 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
117105, 112, 116fsumsub 15825 . . . . . . . . . . . . . 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 14014 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ Fin
11991nn0red 12590 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
120 eluzelz 12889 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
121120adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
122121zred 12724 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
123 prmuz2 16734 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
124123ad2antlr 727 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ (ℤ‘2))
12590nnnn0d 12589 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
126 bernneq3 14271 . . . . . . . . . . . . . . . . . . . . 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 11414 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
129128ord 864 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
13090nnzd 12642 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
131 pcdvdsb 16908 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ0) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13267, 130, 125, 131syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13384, 125nnexpcld 14285 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
134133nnzd 12642 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
135 dvdsle 16348 . . . . . . . . . . . . . . . . . . . . . . . 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 12282 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
138137, 108lenltd 11408 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
139136, 138sylibd 239 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
140132, 139sylbid 240 . . . . . . . . . . . . . . . . . . . . 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 12892 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
144143adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
145119, 108, 122, 142, 144letrd 11419 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
146 eluz 12893 . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))))
149 fzss2 13605 . . . . . . . . . . . . . . . . 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 16935 . . . . . . . . . . . . . . . 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 14386 . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . . 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 15770 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
158105, 116fsumcl 15770 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
159119recnd 11290 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
160157, 158, 159subaddd 11639 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 2752 . . . . . . . . . . 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, 162imbitrid 244 . . . . . . . . . 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 3166 . . . . . . . . 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 412 . . . . . . 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 12715 . . . . 5 (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
169168imp 406 . . . 4 ((𝑁 ∈ ℕ0𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
170 oveq2 7440 . . . . . . 7 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
171170sumeq1d 15737 . . . . . 6 (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
172171eqeq2d 2747 . . . . 5 (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
173172rspcv 3617 . . . 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 1116 . 2 ((𝑀 ∈ (ℤ𝑁) ∧ 𝑁 ∈ ℕ0𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
1761753com12 1123 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 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wss 3950  ifcif 4524   class class class wbr 5142  cfv 6560  (class class class)co 7432  Fincfn 8986  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161   < clt 11296  cle 11297  cmin 11493   / cdiv 11921  cn 12267  2c2 12322  0cn0 12528  cz 12615  cuz 12879  ...cfz 13548  cfl 13831  cexp 14103  !cfa 14313  chash 14370  Σcsu 15723  cdvds 16291  cprime 16709   pCnt cpc 16875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-fz 13549  df-fzo 13696  df-fl 13833  df-mod 13911  df-seq 14044  df-exp 14104  df-fac 14314  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724  df-dvds 16292  df-gcd 16533  df-prm 16710  df-pc 16876
This theorem is referenced by:  pcbc  16939
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