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Theorem pcfac 15797
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 6344 . . . . . . . 8 (𝑥 = 0 → (ℤ𝑥) = (ℤ‘0))
2 fveq2 6344 . . . . . . . . . 10 (𝑥 = 0 → (!‘𝑥) = (!‘0))
32oveq2d 6821 . . . . . . . . 9 (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4 oveq1 6812 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥 / (𝑃𝑘)) = (0 / (𝑃𝑘)))
54fveq2d 6348 . . . . . . . . . 10 (𝑥 = 0 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(0 / (𝑃𝑘))))
65sumeq2sdv 14626 . . . . . . . . 9 (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
73, 6eqeq12d 2767 . . . . . . . 8 (𝑥 = 0 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘)))))
81, 7raleqbidv 3283 . . . . . . 7 (𝑥 = 0 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘)))))
98imbi2d 329 . . . . . 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))))))
10 fveq2 6344 . . . . . . . 8 (𝑥 = 𝑛 → (ℤ𝑥) = (ℤ𝑛))
11 fveq2 6344 . . . . . . . . . 10 (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
1211oveq2d 6821 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
13 oveq1 6812 . . . . . . . . . . 11 (𝑥 = 𝑛 → (𝑥 / (𝑃𝑘)) = (𝑛 / (𝑃𝑘)))
1413fveq2d 6348 . . . . . . . . . 10 (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑛 / (𝑃𝑘))))
1514sumeq2sdv 14626 . . . . . . . . 9 (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))))
1612, 15eqeq12d 2767 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
1710, 16raleqbidv 3283 . . . . . . 7 (𝑥 = 𝑛 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
1817imbi2d 329 . . . . . 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))))))
19 fveq2 6344 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (ℤ𝑥) = (ℤ‘(𝑛 + 1)))
20 fveq2 6344 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
2120oveq2d 6821 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
22 oveq1 6812 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝑥 / (𝑃𝑘)) = ((𝑛 + 1) / (𝑃𝑘)))
2322fveq2d 6348 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘((𝑛 + 1) / (𝑃𝑘))))
2423sumeq2sdv 14626 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))
2521, 24eqeq12d 2767 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
2619, 25raleqbidv 3283 . . . . . . 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)))))
2726imbi2d 329 . . . . . 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))))))
28 fveq2 6344 . . . . . . . 8 (𝑥 = 𝑁 → (ℤ𝑥) = (ℤ𝑁))
29 fveq2 6344 . . . . . . . . . 10 (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
3029oveq2d 6821 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
31 oveq1 6812 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑥 / (𝑃𝑘)) = (𝑁 / (𝑃𝑘)))
3231fveq2d 6348 . . . . . . . . . 10 (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑁 / (𝑃𝑘))))
3332sumeq2sdv 14626 . . . . . . . . 9 (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
3430, 33eqeq12d 2767 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3528, 34raleqbidv 3283 . . . . . . 7 (𝑥 = 𝑁 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3635imbi2d 329 . . . . . 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))))))
37 fzfid 12958 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (1...𝑚) ∈ Fin)
38 sumz 14644 . . . . . . . . . 10 (((1...𝑚) ⊆ (ℤ‘1) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
3938olcs 409 . . . . . . . . 9 ((1...𝑚) ∈ Fin → Σ𝑘 ∈ (1...𝑚)0 = 0)
4037, 39syl 17 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → Σ𝑘 ∈ (1...𝑚)0 = 0)
41 0nn0 11491 . . . . . . . . . . 11 0 ∈ ℕ0
4241a1i 11 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ∈ ℕ0)
43 elfznn 12555 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
4443nnnn0d 11535 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
45 nn0uz 11907 . . . . . . . . . . . 12 0 = (ℤ‘0)
4644, 45syl6eleq 2841 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘0))
4746adantl 473 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ (ℤ‘0))
48 simpll 807 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
49 pcfaclem 15796 . . . . . . . . . 10 ((0 ∈ ℕ0𝑘 ∈ (ℤ‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃𝑘))) = 0)
5042, 47, 48, 49syl3anc 1473 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃𝑘))) = 0)
5150sumeq2dv 14624 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
52 fac0 13249 . . . . . . . . . . 11 (!‘0) = 1
5352oveq2i 6816 . . . . . . . . . 10 (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
54 pc1 15754 . . . . . . . . . 10 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
5553, 54syl5eq 2798 . . . . . . . . 9 (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
5655adantr 472 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = 0)
5740, 51, 563eqtr4rd 2797 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
5857ralrimiva 3096 . . . . . 6 (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
59 nn0z 11584 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
6059adantr 472 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
61 uzid 11886 . . . . . . . . . . 11 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
62 peano2uz 11926 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑛) → (𝑛 + 1) ∈ (ℤ𝑛))
6360, 61, 623syl 18 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → (𝑛 + 1) ∈ (ℤ𝑛))
64 uzss 11892 . . . . . . . . . 10 ((𝑛 + 1) ∈ (ℤ𝑛) → (ℤ‘(𝑛 + 1)) ⊆ (ℤ𝑛))
65 ssralv 3799 . . . . . . . . . 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)))))
6663, 64, 653syl 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)))))
67 oveq1 6812 . . . . . . . . . . 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))))
68 simpll 807 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
69 facp1 13251 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
7068, 69syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
7170oveq2d 6821 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
72 simplr 809 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
73 faccl 13256 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
74 nnz 11583 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
75 nnne0 11237 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
7674, 75jca 555 . . . . . . . . . . . . . . 15 ((!‘𝑛) ∈ ℕ → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
7768, 73, 763syl 18 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
78 nn0p1nn 11516 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
79 nnz 11583 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
80 nnne0 11237 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
8179, 80jca 555 . . . . . . . . . . . . . . 15 ((𝑛 + 1) ∈ ℕ → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
8268, 78, 813syl 18 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
83 pcmul 15750 . . . . . . . . . . . . . 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))))
8472, 77, 82, 83syl3anc 1473 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))) = ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))))
8571, 84eqtr2d 2787 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
8668adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ0)
8786nn0zd 11664 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
88 prmnn 15582 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8988ad2antlr 765 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
90 nnexpcl 13059 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ)
9189, 44, 90syl2an 495 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃𝑘) ∈ ℕ)
92 fldivp1 15795 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℤ ∧ (𝑃𝑘) ∈ ℕ) → ((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = if((𝑃𝑘) ∥ (𝑛 + 1), 1, 0))
9387, 91, 92syl2anc 696 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = if((𝑃𝑘) ∥ (𝑛 + 1), 1, 0))
94 elfzuz 12523 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘1))
9568, 78syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ)
9672, 95pccld 15749 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
9796nn0zd 11664 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
98 elfz5 12519 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ (ℤ‘1) ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
9994, 97, 98syl2anr 496 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
100 simpllr 817 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
10186, 78syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℕ)
102101nnzd 11665 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
10344adantl 473 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ0)
104 pcdvdsb 15767 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
105100, 102, 103, 104syl3anc 1473 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
10699, 105bitr2d 269 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑃𝑘) ∥ (𝑛 + 1) ↔ 𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1)))))
107106ifbid 4244 . . . . . . . . . . . . . . . 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))
10893, 107eqtrd 2786 . . . . . . . . . . . . . . 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))
109108sumeq2dv 14624 . . . . . . . . . . . . . 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))
110 fzfid 12958 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
11168nn0red 11536 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
112 peano2re 10393 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
113111, 112syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℝ)
114113adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℝ)
115114, 91nndivred 11253 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃𝑘)) ∈ ℝ)
116115flcld 12785 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℤ)
117116zcnd 11667 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
118111adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℝ)
119118, 91nndivred 11253 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃𝑘)) ∈ ℝ)
120119flcld 12785 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℤ)
121120zcnd 11667 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
122110, 117, 121fsumsub 14711 . . . . . . . . . . . . . 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)))))
123 fzfi 12957 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ Fin
12496nn0red 11536 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
125 eluzelz 11881 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
126125adantl 473 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
127126zred 11666 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
128 prmuz2 15602 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
129128ad2antlr 765 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ (ℤ‘2))
13095nnnn0d 11535 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
131 bernneq3 13178 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ (ℤ‘2) ∧ (𝑛 + 1) ∈ ℕ0) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
132129, 130, 131syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
133124, 113letrid 10373 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
134133ord 391 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
13595nnzd 11665 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
136 pcdvdsb 15767 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ0) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13772, 135, 130, 136syl3anc 1473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13889, 130nnexpcld 13216 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
139138nnzd 11665 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
140 dvdsle 15226 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
141139, 95, 140syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
142138nnred 11219 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
143142, 113lenltd 10367 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
144141, 143sylibd 229 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
145137, 144sylbid 230 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
146134, 145syld 47 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
147132, 146mt4d 152 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
148 eluzle 11884 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
149148adantl 473 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
150124, 113, 127, 147, 149letrd 10378 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
151 eluz 11885 . . . . . . . . . . . . . . . . . . 19 (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
15297, 126, 151syl2anc 696 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
153150, 152mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))))
154 fzss2 12566 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
155153, 154syl 17 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
156 sumhash 15794 . . . . . . . . . . . . . . . 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)))))
157123, 155, 156sylancr 698 . . . . . . . . . . . . . . 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)))))
158 hashfz1 13320 . . . . . . . . . . . . . . . 16 ((𝑃 pCnt (𝑛 + 1)) ∈ ℕ0 → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
15996, 158syl 17 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
160157, 159eqtrd 2786 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (𝑃 pCnt (𝑛 + 1)))
161109, 122, 1603eqtr3d 2794 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))) = (𝑃 pCnt (𝑛 + 1)))
162110, 117fsumcl 14655 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
163110, 121fsumcl 14655 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
164124recnd 10252 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
165162, 163, 164subaddd 10594 . . . . . . . . . . . . 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)))))
166161, 165mpbid 222 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) + (𝑃 pCnt (𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))
16785, 166eqeq12d 2767 . . . . . . . . . . 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)))))
16867, 167syl5ib 234 . . . . . . . . . 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)))))
169168ralimdva 3092 . . . . . . . . 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)))))
17066, 169syld 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)))))
171170ex 449 . . . . . . 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))))))
172171a2d 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))))))
1739, 18, 27, 36, 58, 172nn0ind 11656 . . . . 5 (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
174173imp 444 . . . 4 ((𝑁 ∈ ℕ0𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
175 oveq2 6813 . . . . . . 7 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
176175sumeq1d 14622 . . . . . 6 (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
177176eqeq2d 2762 . . . . 5 (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
178177rspcv 3437 . . . 4 (𝑀 ∈ (ℤ𝑁) → (∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
179174, 178syl5 34 . . 3 (𝑀 ∈ (ℤ𝑁) → ((𝑁 ∈ ℕ0𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
1801793impib 1108 . 2 ((𝑀 ∈ (ℤ𝑁) ∧ 𝑁 ∈ ℕ0𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
1811803com12 1117 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 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wss 3707  ifcif 4222   class class class wbr 4796  cfv 6041  (class class class)co 6805  Fincfn 8113  cr 10119  0cc0 10120  1c1 10121   + caddc 10123   · cmul 10125   < clt 10258  cle 10259  cmin 10450   / cdiv 10868  cn 11204  2c2 11254  0cn0 11476  cz 11561  cuz 11871  ...cfz 12511  cfl 12777  cexp 13046  !cfa 13246  chash 13303  Σcsu 14607  cdvds 15174  cprime 15579   pCnt cpc 15735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-fz 12512  df-fzo 12652  df-fl 12779  df-mod 12855  df-seq 12988  df-exp 13047  df-fac 13247  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-sum 14608  df-dvds 15175  df-gcd 15411  df-prm 15580  df-pc 15736
This theorem is referenced by:  pcbc  15798
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