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Theorem pcfac 12331
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 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5511 . . . . . . . 8 (𝑥 = 0 → (ℤ𝑥) = (ℤ‘0))
2 fveq2 5511 . . . . . . . . . 10 (𝑥 = 0 → (!‘𝑥) = (!‘0))
32oveq2d 5885 . . . . . . . . 9 (𝑥 = 0 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘0)))
4 fvoveq1 5892 . . . . . . . . . 10 (𝑥 = 0 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(0 / (𝑃𝑘))))
54sumeq2sdv 11362 . . . . . . . . 9 (𝑥 = 0 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
63, 5eqeq12d 2192 . . . . . . . 8 (𝑥 = 0 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘)))))
71, 6raleqbidv 2684 . . . . . . 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 230 . . . . . 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 5511 . . . . . . . 8 (𝑥 = 𝑛 → (ℤ𝑥) = (ℤ𝑛))
10 fveq2 5511 . . . . . . . . . 10 (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛))
1110oveq2d 5885 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑛)))
12 fvoveq1 5892 . . . . . . . . . 10 (𝑥 = 𝑛 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑛 / (𝑃𝑘))))
1312sumeq2sdv 11362 . . . . . . . . 9 (𝑥 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))))
1411, 13eqeq12d 2192 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
159, 14raleqbidv 2684 . . . . . . 7 (𝑥 = 𝑛 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑛)(𝑃 pCnt (!‘𝑛)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))))
1615imbi2d 230 . . . . . 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 5511 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (ℤ𝑥) = (ℤ‘(𝑛 + 1)))
18 fveq2 5511 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (!‘𝑥) = (!‘(𝑛 + 1)))
1918oveq2d 5885 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘(𝑛 + 1))))
20 fvoveq1 5892 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘((𝑛 + 1) / (𝑃𝑘))))
2120sumeq2sdv 11362 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))
2219, 21eqeq12d 2192 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘(𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘)))))
2317, 22raleqbidv 2684 . . . . . . 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 230 . . . . . 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 5511 . . . . . . . 8 (𝑥 = 𝑁 → (ℤ𝑥) = (ℤ𝑁))
26 fveq2 5511 . . . . . . . . . 10 (𝑥 = 𝑁 → (!‘𝑥) = (!‘𝑁))
2726oveq2d 5885 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑃 pCnt (!‘𝑥)) = (𝑃 pCnt (!‘𝑁)))
28 fvoveq1 5892 . . . . . . . . . 10 (𝑥 = 𝑁 → (⌊‘(𝑥 / (𝑃𝑘))) = (⌊‘(𝑁 / (𝑃𝑘))))
2928sumeq2sdv 11362 . . . . . . . . 9 (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
3027, 29eqeq12d 2192 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3125, 30raleqbidv 2684 . . . . . . 7 (𝑥 = 𝑁 → (∀𝑚 ∈ (ℤ𝑥)(𝑃 pCnt (!‘𝑥)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑥 / (𝑃𝑘))) ↔ ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
3231imbi2d 230 . . . . . 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 1zzd 9269 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → 1 ∈ ℤ)
34 eluzelz 9526 . . . . . . . . . . 11 (𝑚 ∈ (ℤ‘0) → 𝑚 ∈ ℤ)
3534adantl 277 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → 𝑚 ∈ ℤ)
3633, 35fzfigd 10417 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (1...𝑚) ∈ Fin)
37 isumz 11381 . . . . . . . . . 10 (((1 ∈ ℤ ∧ (1...𝑚) ⊆ (ℤ‘1) ∧ ∀𝑗 ∈ (ℤ‘1)DECID 𝑗 ∈ (1...𝑚)) ∨ (1...𝑚) ∈ Fin) → Σ𝑘 ∈ (1...𝑚)0 = 0)
3837olcs 736 . . . . . . . . 9 ((1...𝑚) ∈ Fin → Σ𝑘 ∈ (1...𝑚)0 = 0)
3936, 38syl 14 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → Σ𝑘 ∈ (1...𝑚)0 = 0)
40 0nn0 9180 . . . . . . . . . 10 0 ∈ ℕ0
41 elfznn 10040 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
4241nnnn0d 9218 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ0)
43 nn0uz 9551 . . . . . . . . . . . 12 0 = (ℤ‘0)
4442, 43eleqtrdi 2270 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘0))
4544adantl 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 12330 . . . . . . . . . 10 ((0 ∈ ℕ0𝑘 ∈ (ℤ‘0) ∧ 𝑃 ∈ ℙ) → (⌊‘(0 / (𝑃𝑘))) = 0)
4840, 45, 46, 47mp3an2i 1342 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(0 / (𝑃𝑘))) = 0)
4948sumeq2dv 11360 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑚)0)
50 fac0 10692 . . . . . . . . . . 11 (!‘0) = 1
5150oveq2i 5880 . . . . . . . . . 10 (𝑃 pCnt (!‘0)) = (𝑃 pCnt 1)
52 pc1 12288 . . . . . . . . . 10 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
5351, 52eqtrid 2222 . . . . . . . . 9 (𝑃 ∈ ℙ → (𝑃 pCnt (!‘0)) = 0)
5453adantr 276 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = 0)
5539, 49, 543eqtr4rd 2221 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ (ℤ‘0)) → (𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
5655ralrimiva 2550 . . . . . 6 (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ‘0)(𝑃 pCnt (!‘0)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(0 / (𝑃𝑘))))
57 nn0z 9262 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
5857adantr 276 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → 𝑛 ∈ ℤ)
59 uzid 9531 . . . . . . . . . . 11 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
60 peano2uz 9572 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑛) → (𝑛 + 1) ∈ (ℤ𝑛))
6158, 59, 603syl 17 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝑃 ∈ ℙ) → (𝑛 + 1) ∈ (ℤ𝑛))
62 uzss 9537 . . . . . . . . . 10 ((𝑛 + 1) ∈ (ℤ𝑛) → (ℤ‘(𝑛 + 1)) ⊆ (ℤ𝑛))
63 ssralv 3219 . . . . . . . . . 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)))))
6461, 62, 633syl 17 . . . . . . . . 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)))))
65 oveq1 5876 . . . . . . . . . . 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))))
66 simpll 527 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
67 facp1 10694 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
6866, 67syl 14 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (!‘(𝑛 + 1)) = ((!‘𝑛) · (𝑛 + 1)))
6968oveq2d 5885 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (!‘(𝑛 + 1))) = (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))))
70 simplr 528 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℙ)
71 faccl 10699 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ)
72 nnz 9261 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℤ)
73 nnne0 8936 . . . . . . . . . . . . . . . 16 ((!‘𝑛) ∈ ℕ → (!‘𝑛) ≠ 0)
7472, 73jca 306 . . . . . . . . . . . . . . 15 ((!‘𝑛) ∈ ℕ → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
7566, 71, 743syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((!‘𝑛) ∈ ℤ ∧ (!‘𝑛) ≠ 0))
76 nn0p1nn 9204 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
77 nnz 9261 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℤ)
78 nnne0 8936 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
7977, 78jca 306 . . . . . . . . . . . . . . 15 ((𝑛 + 1) ∈ ℕ → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
8066, 76, 793syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ≠ 0))
81 pcmul 12284 . . . . . . . . . . . . . 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))))
8270, 75, 80, 81syl3anc 1238 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt ((!‘𝑛) · (𝑛 + 1))) = ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))))
8369, 82eqtr2d 2211 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (!‘𝑛)) + (𝑃 pCnt (𝑛 + 1))) = (𝑃 pCnt (!‘(𝑛 + 1))))
8466adantr 276 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℕ0)
8584nn0zd 9362 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑛 ∈ ℤ)
86 prmnn 12093 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8786ad2antlr 489 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ ℕ)
88 nnexpcl 10519 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑃𝑘) ∈ ℕ)
8987, 42, 88syl2an 289 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑃𝑘) ∈ ℕ)
90 fldivp1 12329 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℤ ∧ (𝑃𝑘) ∈ ℕ) → ((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = if((𝑃𝑘) ∥ (𝑛 + 1), 1, 0))
9185, 89, 90syl2anc 411 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((⌊‘((𝑛 + 1) / (𝑃𝑘))) − (⌊‘(𝑛 / (𝑃𝑘)))) = if((𝑃𝑘) ∥ (𝑛 + 1), 1, 0))
92 elfzuz 10007 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ (ℤ‘1))
9366, 76syl 14 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ)
9470, 93pccld 12283 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
9594nn0zd 9362 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
96 elfz5 10003 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ (ℤ‘1) ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
9792, 95, 96syl2anr 290 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))) ↔ 𝑘 ≤ (𝑃 pCnt (𝑛 + 1))))
98 simpllr 534 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑃 ∈ ℙ)
9984, 76syl 14 . . . . . . . . . . . . . . . . . . . 20 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℕ)
10099nnzd 9363 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 + 1) ∈ ℤ)
10142adantl 277 . . . . . . . . . . . . . . . . . . 19 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ0)
102 pcdvdsb 12302 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
10398, 100, 101, 102syl3anc 1238 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑘 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑘) ∥ (𝑛 + 1)))
10497, 103bitr2d 189 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑃𝑘) ∥ (𝑛 + 1) ↔ 𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1)))))
105104ifbid 3555 . . . . . . . . . . . . . . . 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))
10691, 105eqtrd 2210 . . . . . . . . . . . . . . 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))
107106sumeq2dv 11360 . . . . . . . . . . . . . 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))
108 1zzd 9269 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 1 ∈ ℤ)
109 eluzelz 9526 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → 𝑚 ∈ ℤ)
110109adantl 277 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℤ)
111108, 110fzfigd 10417 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (1...𝑚) ∈ Fin)
112 znq 9613 . . . . . . . . . . . . . . . . . 18 (((𝑛 + 1) ∈ ℤ ∧ (𝑃𝑘) ∈ ℕ) → ((𝑛 + 1) / (𝑃𝑘)) ∈ ℚ)
113100, 89, 112syl2anc 411 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑛 + 1) / (𝑃𝑘)) ∈ ℚ)
114113flqcld 10263 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℤ)
115114zcnd 9365 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
116 znq 9613 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℤ ∧ (𝑃𝑘) ∈ ℕ) → (𝑛 / (𝑃𝑘)) ∈ ℚ)
11785, 89, 116syl2anc 411 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (𝑛 / (𝑃𝑘)) ∈ ℚ)
118117flqcld 10263 . . . . . . . . . . . . . . . 16 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℤ)
119118zcnd 9365 . . . . . . . . . . . . . . 15 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑘 ∈ (1...𝑚)) → (⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
120111, 115, 119fsumsub 11444 . . . . . . . . . . . . . 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)))))
12194nn0red 9219 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
12266nn0red 9219 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑛 ∈ ℝ)
123 peano2re 8083 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
124122, 123syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℝ)
125110zred 9364 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ ℝ)
12693nnzd 9363 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℤ)
127 zdcle 9318 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℤ) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
12895, 126, 127syl2anc 411 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → DECID (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
129 zletric 9286 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℤ) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
13095, 126, 129syl2anc 411 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) ∨ (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
131130ord 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → (𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1))))
13293nnnn0d 9218 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
133 pcdvdsb 12302 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ0) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13470, 126, 132, 133syl3anc 1238 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1)))
13587, 132nnexpcld 10661 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℕ)
136135nnzd 9363 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℤ)
137 dvdsle 11833 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃↑(𝑛 + 1)) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
138136, 93, 137syl2anc 411 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → (𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1)))
139135nnred 8921 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃↑(𝑛 + 1)) ∈ ℝ)
140139, 124lenltd 8065 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
141138, 140sylibd 149 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑃↑(𝑛 + 1)) ∥ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
142134, 141sylbid 150 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ((𝑛 + 1) ≤ (𝑃 pCnt (𝑛 + 1)) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
143131, 142syld 45 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (¬ (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1) → ¬ (𝑛 + 1) < (𝑃↑(𝑛 + 1))))
144 prmuz2 12114 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
145144ad2antlr 489 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑃 ∈ (ℤ‘2))
146 bernneq3 10628 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ (ℤ‘2) ∧ (𝑛 + 1) ∈ ℕ0) → (𝑛 + 1) < (𝑃↑(𝑛 + 1)))
147145, 132, 146syl2anc 411 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ ℕ0𝑃 ∈ 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))))
149128, 143, 147, 148syl3c 63 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ (𝑛 + 1))
150 eluzle 9529 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → (𝑛 + 1) ≤ 𝑚)
151150adantl 277 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑛 + 1) ≤ 𝑚)
152121, 124, 125, 149, 151letrd 8071 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚)
153 eluz 9530 . . . . . . . . . . . . . . . . . . 19 (((𝑃 pCnt (𝑛 + 1)) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
15495, 110, 153syl2anc 411 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) ↔ (𝑃 pCnt (𝑛 + 1)) ≤ 𝑚))
155152, 154mpbird 167 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → 𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))))
156 fzss2 10050 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (ℤ‘(𝑃 pCnt (𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
157155, 156syl 14 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (1...(𝑃 pCnt (𝑛 + 1))) ⊆ (1...𝑚))
158 elfzelz 10011 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℤ)
159158adantl 277 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℤ)
160 1zzd 9269 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
16195adantr 276 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → (𝑃 pCnt (𝑛 + 1)) ∈ ℤ)
162 fzdcel 10026 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝑃 pCnt (𝑛 + 1)) ∈ ℤ) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
163159, 160, 161, 162syl3anc 1238 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) ∧ 𝑗 ∈ (1...𝑚)) → DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
164163ralrimiva 2550 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → ∀𝑗 ∈ (1...𝑚)DECID 𝑗 ∈ (1...(𝑃 pCnt (𝑛 + 1))))
165 sumhashdc 12328 . . . . . . . . . . . . . . . 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)))))
166111, 157, 164, 165syl3anc 1238 . . . . . . . . . . . . . . 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)))))
167 hashfz1 10747 . . . . . . . . . . . . . . . 16 ((𝑃 pCnt (𝑛 + 1)) ∈ ℕ0 → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
16894, 167syl 14 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (♯‘(1...(𝑃 pCnt (𝑛 + 1)))) = (𝑃 pCnt (𝑛 + 1)))
169166, 168eqtrd 2210 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)if(𝑘 ∈ (1...(𝑃 pCnt (𝑛 + 1))), 1, 0) = (𝑃 pCnt (𝑛 + 1)))
170107, 120, 1693eqtr3d 2218 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) − Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘)))) = (𝑃 pCnt (𝑛 + 1)))
171111, 115fsumcl 11392 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))) ∈ ℂ)
172111, 119fsumcl 11392 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) ∈ ℂ)
17394nn0cnd 9220 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
174171, 172, 173subaddd 8276 . . . . . . . . . . . . 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)))))
175170, 174mpbid 147 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑃 ∈ ℙ) ∧ 𝑚 ∈ (ℤ‘(𝑛 + 1))) → (Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑛 / (𝑃𝑘))) + (𝑃 pCnt (𝑛 + 1))) = Σ𝑘 ∈ (1...𝑚)(⌊‘((𝑛 + 1) / (𝑃𝑘))))
17683, 175eqeq12d 2192 . . . . . . . . . . 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)))))
17765, 176imbitrid 154 . . . . . . . . . 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)))))
178177ralimdva 2544 . . . . . . . . 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)))))
17964, 178syld 45 . . . . . . . 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)))))
180179ex 115 . . . . . . 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))))))
181180a2d 26 . . . . . 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))))))
1828, 16, 24, 32, 56, 181nn0ind 9356 . . . . 5 (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘)))))
183182imp 124 . . . 4 ((𝑁 ∈ ℕ0𝑃 ∈ ℙ) → ∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))))
184 oveq2 5877 . . . . . . 7 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
185184sumeq1d 11358 . . . . . 6 (𝑚 = 𝑀 → Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
186185eqeq2d 2189 . . . . 5 (𝑚 = 𝑀 → ((𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) ↔ (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
187186rspcv 2837 . . . 4 (𝑀 ∈ (ℤ𝑁) → (∀𝑚 ∈ (ℤ𝑁)(𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑚)(⌊‘(𝑁 / (𝑃𝑘))) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
188183, 187syl5 32 . . 3 (𝑀 ∈ (ℤ𝑁) → ((𝑁 ∈ ℕ0𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘)))))
1891883impib 1201 . 2 ((𝑀 ∈ (ℤ𝑁) ∧ 𝑁 ∈ ℕ0𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
1901893com12 1207 1 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))
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 3129  ifcif 3534   class class class wbr 4000  cfv 5212  (class class class)co 5869  Fincfn 6734  cr 7801  0cc0 7802  1c1 7803   + caddc 7805   · cmul 7807   < clt 7982  cle 7983  cmin 8118   / cdiv 8618  cn 8908  2c2 8959  0cn0 9165  cz 9242  cuz 9517  cq 9608  ...cfz 9995  cfl 10254  cexp 10505  !cfa 10689  chash 10739  Σcsu 11345  cdvds 11778  cprime 12090   pCnt cpc 12267
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-2o 6412  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-dvds 11779  df-gcd 11927  df-prm 12091  df-pc 12268
This theorem is referenced by:  pcbc  12332
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