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Theorem pcbc 16304

Description: Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

**Assertion**
Ref	Expression
pcbc	⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃↑𝑘))) − ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))))

Distinct variable groups: 𝑃,𝑘 𝑘,𝑁 𝑘,𝐾

**Proof of Theorem pcbc**
Step	Hyp	Ref	Expression
1		simp3 1135	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ)
2		nnnn0 11954	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3	2	3ad2ant1 1130	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℕ₀)
4	3	faccld 13707	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈ ℕ)
5	4	nnzd 12138	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈ ℤ)
6	4	nnne0d 11737	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ≠ 0)
7		fznn0sub 13001	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
8	7	3ad2ant2 1131	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℕ₀)
9	8	faccld 13707	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
10		elfznn0 13062	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
11	10	3ad2ant2 1131	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℕ₀)
12	11	faccld 13707	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈ ℕ)
13	9, 12	nnmulcld 11740	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ)
14		pcdiv 16257	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ ((!‘𝑁) ∈ ℤ ∧ (!‘𝑁) ≠ 0) ∧ ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) → (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
15	1, 5, 6, 13, 14	syl121anc 1372	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
16		bcval2 13728	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
17	16	3ad2ant2 1131	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
18	17	oveq2d 7172	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
19		fzfid 13403	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (1...𝑁) ∈ Fin)
20		nnre 11694	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
21	20	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℝ)
22	21	adantr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
23		simpl3 1190	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑃 ∈ ℙ)
24		prmnn 16083	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑃 ∈ ℕ)
26		elfznn 12998	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
27	26	nnnn0d 12007	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ₀)
28	27	adantl 485	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ₀)
29	25, 28	nnexpcld 13669	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑃↑𝑘) ∈ ℕ)
30	22, 29	nndivred 11741	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
31	30	flcld 13230	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ)
32	31	zcnd 12140	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℂ)
33	11	nn0red 12008	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℝ)
34	21, 33	resubcld 11119	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℝ)
35	34	adantr 484	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 − 𝐾) ∈ ℝ)
36	35, 29	nndivred 11741	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝑁 − 𝐾) / (𝑃↑𝑘)) ∈ ℝ)
37	36	flcld 13230	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℤ)
38	37	zcnd 12140	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℂ)
39	33	adantr 484	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝐾 ∈ ℝ)
40	39, 29	nndivred 11741	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝐾 / (𝑃↑𝑘)) ∈ ℝ)
41	40	flcld 13230	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℤ)
42	41	zcnd 12140	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℂ)
43	38, 42	addcld 10711	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))) ∈ ℂ)
44	19, 32, 43	fsumsub 15204	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃↑𝑘))) − ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))) − Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))))
45	3	nn0zd 12137	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℤ)
46		uzid 12310	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
47	45, 46	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘𝑁))
48		pcfac 16303	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))))
49	3, 47, 1, 48	syl3anc 1368	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))))
50	11	nn0ge0d 12010	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 0 ≤ 𝐾)
51	21, 33	subge02d 11283	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁))
52	50, 51	mpbid 235	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ≤ 𝑁)
53	11	nn0zd 12137	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℤ)
54	45, 53	zsubcld 12144	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℤ)
55		eluz 12309	. . . . . . . . 9 ⊢ (((𝑁 − 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)) ↔ (𝑁 − 𝐾) ≤ 𝑁))
56	54, 45, 55	syl2anc 587	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)) ↔ (𝑁 − 𝐾) ≤ 𝑁))
57	52, 56	mpbird 260	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)))
58		pcfac 16303	. . . . . . 7 ⊢ (((𝑁 − 𝐾) ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘(𝑁 − 𝐾))) = Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))))
59	8, 57, 1, 58	syl3anc 1368	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘(𝑁 − 𝐾))) = Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))))
60		elfzuz3 12966	. . . . . . . 8 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
61	60	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘𝐾))
62		pcfac 16303	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝐾) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))
63	11, 61, 1, 62	syl3anc 1368	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))
64	59, 63	oveq12d 7174	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((𝑃 pCnt (!‘(𝑁 − 𝐾))) + (𝑃 pCnt (!‘𝐾))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘)))))
65	9	nnzd 12138	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℤ)
66	9	nnne0d 11737	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ≠ 0)
67	12	nnzd 12138	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈ ℤ)
68	12	nnne0d 11737	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ≠ 0)
69		pcmul 16256	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ((!‘(𝑁 − 𝐾)) ∈ ℤ ∧ (!‘(𝑁 − 𝐾)) ≠ 0) ∧ ((!‘𝐾) ∈ ℤ ∧ (!‘𝐾) ≠ 0)) → (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) = ((𝑃 pCnt (!‘(𝑁 − 𝐾))) + (𝑃 pCnt (!‘𝐾))))
70	1, 65, 66, 67, 68, 69	syl122anc 1376	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) = ((𝑃 pCnt (!‘(𝑁 − 𝐾))) + (𝑃 pCnt (!‘𝐾))))
71	19, 38, 42	fsumadd 15157	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘)))))
72	64, 70, 71	3eqtr4d 2803	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) = Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))))
73	49, 72	oveq12d 7174	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))) − Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))))
74	44, 73	eqtr4d 2796	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃↑𝑘))) − ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
75	15, 18, 74	3eqtr4d 2803	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃↑𝑘))) − ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 class class class wbr 5036 ‘cfv 6340 (class class class)co 7156 ℝcr 10587 0cc0 10588 1c1 10589 + caddc 10591 · cmul 10593 ≤ cle 10727 − cmin 10921 / cdiv 11348 ℕcn 11687 ℕ₀cn0 11947 ℤcz 12033 ℤ_≥cuz 12295 ...cfz 12952 ⌊cfl 13222 ↑cexp 13492 !cfa 13696 Ccbc 13725 Σcsu 15103 ℙcprime 16080 pCnt cpc 16241

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-2o 8119 df-er 8305 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-sup 8952 df-inf 8953 df-oi 9020 df-card 9414 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-n0 11948 df-z 12034 df-uz 12296 df-q 12402 df-rp 12444 df-fz 12953 df-fzo 13096 df-fl 13224 df-mod 13300 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-clim 14906 df-sum 15104 df-dvds 15669 df-gcd 15907 df-prm 16081 df-pc 16242

This theorem is referenced by: pcbcctr 25972

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