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Theorem pcbc 16862

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 1136	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ)
2		nnnn0 12503	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3	2	3ad2ant1 1131	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℕ₀)
4	3	faccld 14269	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈ ℕ)
5	4	nnzd 12609	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈ ℤ)
6	4	nnne0d 12286	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ≠ 0)
7		fznn0sub 13559	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
8	7	3ad2ant2 1132	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℕ₀)
9	8	faccld 14269	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
10		elfznn0 13620	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
11	10	3ad2ant2 1132	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℕ₀)
12	11	faccld 14269	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈ ℕ)
13	9, 12	nnmulcld 12289	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ)
14		pcdiv 16814	. . 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 1373	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
16		bcval2 14290	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
17	16	3ad2ant2 1132	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
18	17	oveq2d 7430	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
19		fzfid 13964	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (1...𝑁) ∈ Fin)
20		nnre 12243	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
21	20	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℝ)
22	21	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
23		simpl3 1191	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑃 ∈ ℙ)
24		prmnn 16638	. . . . . . . . 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 13556	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
27	26	nnnn0d 12556	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ₀)
28	27	adantl 481	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ₀)
29	25, 28	nnexpcld 14233	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑃↑𝑘) ∈ ℕ)
30	22, 29	nndivred 12290	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
31	30	flcld 13789	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ)
32	31	zcnd 12691	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℂ)
33	11	nn0red 12557	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℝ)
34	21, 33	resubcld 11666	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℝ)
35	34	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 − 𝐾) ∈ ℝ)
36	35, 29	nndivred 12290	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝑁 − 𝐾) / (𝑃↑𝑘)) ∈ ℝ)
37	36	flcld 13789	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℤ)
38	37	zcnd 12691	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℂ)
39	33	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝐾 ∈ ℝ)
40	39, 29	nndivred 12290	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝐾 / (𝑃↑𝑘)) ∈ ℝ)
41	40	flcld 13789	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℤ)
42	41	zcnd 12691	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℂ)
43	38, 42	addcld 11257	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))) ∈ ℂ)
44	19, 32, 43	fsumsub 15760	. . 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 12608	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℤ)
46		uzid 12861	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
47	45, 46	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘𝑁))
48		pcfac 16861	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))))
49	3, 47, 1, 48	syl3anc 1369	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))))
50	11	nn0ge0d 12559	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 0 ≤ 𝐾)
51	21, 33	subge02d 11830	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁))
52	50, 51	mpbid 231	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ≤ 𝑁)
53	11	nn0zd 12608	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℤ)
54	45, 53	zsubcld 12695	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℤ)
55		eluz 12860	. . . . . . . . 9 ⊢ (((𝑁 − 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)) ↔ (𝑁 − 𝐾) ≤ 𝑁))
56	54, 45, 55	syl2anc 583	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)) ↔ (𝑁 − 𝐾) ≤ 𝑁))
57	52, 56	mpbird 257	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)))
58		pcfac 16861	. . . . . . 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 1369	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘(𝑁 − 𝐾))) = Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))))
60		elfzuz3 13524	. . . . . . . 8 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
61	60	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘𝐾))
62		pcfac 16861	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝐾) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))
63	11, 61, 1, 62	syl3anc 1369	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))
64	59, 63	oveq12d 7432	. . . . 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 12609	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℤ)
66	9	nnne0d 12286	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ≠ 0)
67	12	nnzd 12609	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈ ℤ)
68	12	nnne0d 12286	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ≠ 0)
69		pcmul 16813	. . . . . 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 1377	. . . . 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 15712	. . . . 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 2778	. . . 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 7432	. . 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 2771	. 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 2778	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 ≤ cle 11273 − cmin 11468 / cdiv 11895 ℕcn 12236 ℕ₀cn0 12496 ℤcz 12582 ℤ_≥cuz 12846 ...cfz 13510 ⌊cfl 13781 ↑cexp 14052 !cfa 14258 Ccbc 14287 Σcsu 15658 ℙcprime 16635 pCnt cpc 16798

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-fz 13511 df-fzo 13654 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 df-dvds 16225 df-gcd 16463 df-prm 16636 df-pc 16799

This theorem is referenced by: pcbcctr 27202

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