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Theorem pcbc 16961

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 1156	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ)
2		nnnn0 12512	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3	2	3ad2ant1 1151	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℕ₀)
4	3	faccld 14322	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈ ℕ)
5	4	nnzd 12618	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈ ℤ)
6	4	nnne0d 12287	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ≠ 0)
7		fznn0sub 13586	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
8	7	3ad2ant2 1152	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℕ₀)
9	8	faccld 14322	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
10		elfznn0 13650	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
11	10	3ad2ant2 1152	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℕ₀)
12	11	faccld 14322	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈ ℕ)
13	9, 12	nnmulcld 12290	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ)
14		pcdiv 16913	. . 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 1402	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
16		bcval2 14343	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
17	16	3ad2ant2 1152	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
18	17	oveq2d 7428	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))))
19		fzfid 14011	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (1...𝑁) ∈ Fin)
20		nnre 12241	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
21	20	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℝ)
22	21	adantr 485	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
23		simpl3 1212	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑃 ∈ ℙ)
24		prmnn 16733	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
25	23, 24	syl 18	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑃 ∈ ℕ)
26		elfznn 13583	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
27	26	nnnn0d 12566	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ₀)
28	27	adantl 486	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ₀)
29	25, 28	nnexpcld 14283	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑃↑𝑘) ∈ ℕ)
30	22, 29	nndivred 12291	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
31	30	flcld 13833	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ)
32	31	zcnd 12702	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℂ)
33	11	nn0red 12567	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℝ)
34	21, 33	resubcld 11643	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℝ)
35	34	adantr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 − 𝐾) ∈ ℝ)
36	35, 29	nndivred 12291	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝑁 − 𝐾) / (𝑃↑𝑘)) ∈ ℝ)
37	36	flcld 13833	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℤ)
38	37	zcnd 12702	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℂ)
39	33	adantr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝐾 ∈ ℝ)
40	39, 29	nndivred 12291	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝐾 / (𝑃↑𝑘)) ∈ ℝ)
41	40	flcld 13833	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℤ)
42	41	zcnd 12702	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℂ)
43	38, 42	addcld 11229	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))) ∈ ℂ)
44	19, 32, 43	fsumsub 15841	. . 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 12617	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℤ)
46		uzid 12878	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
47	45, 46	syl 18	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘𝑁))
48		pcfac 16960	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))))
49	3, 47, 1, 48	syl3anc 1398	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))))
50	11	nn0ge0d 12569	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 0 ≤ 𝐾)
51	21, 33	subge02d 11807	. . . . . . . . 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 12617	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℤ)
54	45, 53	zsubcld 12706	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℤ)
55		eluz 12877	. . . . . . . . 9 ⊢ (((𝑁 − 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝑁 − 𝐾)) ↔ (𝑁 − 𝐾) ≤ 𝑁))
56	54, 45, 55	syl2anc 595	. . . . . . . 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 16960	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘(𝑁 − 𝐾))) = Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))))
60		elfzuz3 13550	. . . . . . . 8 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐾))
61	60	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ_≥‘𝐾))
62		pcfac 16960	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝐾) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))
63	11, 61, 1, 62	syl3anc 1398	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))
64	59, 63	oveq12d 7430	. . . . 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 12618	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℤ)
66	9	nnne0d 12287	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ≠ 0)
67	12	nnzd 12618	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈ ℤ)
68	12	nnne0d 12287	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ≠ 0)
69		pcmul 16912	. . . . . 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 1406	. . . . 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 15793	. . . . 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 2808	. . . 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 7430	. . 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 2801	. 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 2808	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 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5110 ‘cfv 6538 (class class class)co 7412 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 ≤ cle 11245 − cmin 11442 / cdiv 11872 ℕcn 12234 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 ...cfz 13536 ⌊cfl 13825 ↑cexp 14099 !cfa 14311 Ccbc 14340 Σcsu 15739 ℙcprime 16730 pCnt cpc 16897

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-q 12974 df-rp 13018 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-dvds 16312 df-gcd 16554 df-prm 16731 df-pc 16898

This theorem is referenced by: pcbcctr 27421

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