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Theorem birthdaylem2 26922

Description: For general 𝑁 and 𝐾, count the fraction of injective functions from 1...𝐾 to 1...𝑁. (Contributed by Mario Carneiro, 7-May-2015.)

**Hypotheses**
Ref	Expression
birthday.s	⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}
birthday.t	⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}

**Assertion**
Ref	Expression
birthdaylem2	⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = (exp‘Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁)))))

Distinct variable groups: 𝑓,𝑘,𝐾 𝑓,𝑁,𝑘 Allowed substitution hints: 𝑆(𝑓,𝑘) 𝑇(𝑓,𝑘)

Proof of Theorem birthdaylem2 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		birthday.t	. . . . . . 7 ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}
2	1	fveq2i 6838	. . . . . 6 ⊢ (♯‘𝑇) = (♯‘{𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)})
3		fzfi 13899	. . . . . . 7 ⊢ (1...𝐾) ∈ Fin
4		fzfi 13899	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
5		hashf1 14384	. . . . . . 7 ⊢ (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘{𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))))
6	3, 4, 5	mp2an 693	. . . . . 6 ⊢ (♯‘{𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
7	2, 6	eqtri 2760	. . . . 5 ⊢ (♯‘𝑇) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
8		elfznn0 13540	. . . . . . . . 9 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ₀)
10		hashfz1 14273	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ₀ → (♯‘(1...𝐾)) = 𝐾)
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = 𝐾)
12	11	fveq2d 6839	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(♯‘(1...𝐾))) = (!‘𝐾))
13		nnnn0 12412	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 14273	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (♯‘(1...𝑁)) = 𝑁)
16	15	adantr 480	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝑁)) = 𝑁)
17	16, 11	oveq12d 7378	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))C(♯‘(1...𝐾))) = (𝑁C𝐾))
18	12, 17	oveq12d 7378	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))) = ((!‘𝐾) · (𝑁C𝐾)))
19	7, 18	eqtrid 2784	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = ((!‘𝐾) · (𝑁C𝐾)))
20	13	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
21	20	faccld 14211	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
22	21	nncnd 12165	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
23		fznn0sub 13476	. . . . . . . . . 10 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
24	23	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℕ₀)
25	24	faccld 14211	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
26	25	nncnd 12165	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
27	25	nnne0d 12199	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ≠ 0)
28	22, 26, 27	divcld 11921	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁 − 𝐾))) ∈ ℂ)
29	9	faccld 14211	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
30	29	nncnd 12165	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
31	29	nnne0d 12199	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ≠ 0)
32	28, 30, 31	divcan2d 11923	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾))) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
33		bcval2 14232	. . . . . . . 8 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
34	33	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
35	22, 26, 30, 27, 31	divdiv1d 11952	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
36	34, 35	eqtr4d 2775	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
37	36	oveq2d 7376	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾))))
38		fzfid 13900	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
39		elfznn 13473	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
40	39	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
41		nnrp 12921	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
42	41	relogcld 26592	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℝ)
43	42	recnd 11164	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℂ)
44	40, 43	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℂ)
45	38, 44	fsumcl 15660	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ)
46		fzfid 13900	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...(𝑁 − 𝐾)) ∈ Fin)
47		elfznn 13473	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(𝑁 − 𝐾)) → 𝑛 ∈ ℕ)
48	47	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...(𝑁 − 𝐾))) → 𝑛 ∈ ℕ)
49	48, 43	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...(𝑁 − 𝐾))) → (log‘𝑛) ∈ ℂ)
50	46, 49	fsumcl 15660	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛) ∈ ℂ)
51		efsub 16029	. . . . . . 7 ⊢ ((Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ ∧ Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛) ∈ ℂ) → (exp‘(Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))))
52	45, 50, 51	syl2anc 585	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))))
53	24	nn0red 12467	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℝ)
54	53	ltp1d 12076	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) < ((𝑁 − 𝐾) + 1))
55		fzdisj 13471	. . . . . . . . . . 11 ⊢ ((𝑁 − 𝐾) < ((𝑁 − 𝐾) + 1) → ((1...(𝑁 − 𝐾)) ∩ (((𝑁 − 𝐾) + 1)...𝑁)) = ∅)
56	54, 55	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1...(𝑁 − 𝐾)) ∩ (((𝑁 − 𝐾) + 1)...𝑁)) = ∅)
57		fznn0sub2 13555	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁))
58	57	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ (0...𝑁))
59		elfzle2 13448	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 𝐾) ∈ (0...𝑁) → (𝑁 − 𝐾) ≤ 𝑁)
60	58, 59	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ≤ 𝑁)
61	60	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ≤ 𝑁)
62		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ∈ ℕ)
63		nnuz 12794	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
64	62, 63	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ∈ (ℤ_≥‘1))
65		nnz 12513	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
66	65	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → 𝑁 ∈ ℤ)
67		elfz5 13436	. . . . . . . . . . . . . 14 ⊢ (((𝑁 − 𝐾) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝐾) ∈ (1...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁))
68	64, 66, 67	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → ((𝑁 − 𝐾) ∈ (1...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁))
69	61, 68	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ∈ (1...𝑁))
70		fzsplit 13470	. . . . . . . . . . . 12 ⊢ ((𝑁 − 𝐾) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
71	69, 70	syl 17	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
72		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (𝑁 − 𝐾) = 0)
73	72	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...(𝑁 − 𝐾)) = (1...0))
74		fz10 13465	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
75	73, 74	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...(𝑁 − 𝐾)) = ∅)
76	75	uneq1d 4120	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
77		uncom 4111	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = ((((𝑁 − 𝐾) + 1)...𝑁) ∪ ∅)
78		un0 4347	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 − 𝐾) + 1)...𝑁) ∪ ∅) = (((𝑁 − 𝐾) + 1)...𝑁)
79	77, 78	eqtri 2760	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (((𝑁 − 𝐾) + 1)...𝑁)
80	72	oveq1d 7375	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((𝑁 − 𝐾) + 1) = (0 + 1))
81		1e0p1 12653	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
82	80, 81	eqtr4di 2790	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((𝑁 − 𝐾) + 1) = 1)
83	82	oveq1d 7375	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (((𝑁 − 𝐾) + 1)...𝑁) = (1...𝑁))
84	79, 83	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (1...𝑁))
85	76, 84	eqtr2d 2773	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
86		elnn0 12407	. . . . . . . . . . . 12 ⊢ ((𝑁 − 𝐾) ∈ ℕ₀ ↔ ((𝑁 − 𝐾) ∈ ℕ ∨ (𝑁 − 𝐾) = 0))
87	24, 86	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝐾) ∈ ℕ ∨ (𝑁 − 𝐾) = 0))
88	71, 85, 87	mpjaodan 961	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
89	56, 88, 38, 44	fsumsplit 15668	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)))
90	89	oveq1d 7375	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)) = ((Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) − Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
91		fzfid 13900	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁 − 𝐾) + 1)...𝑁) ∈ Fin)
92		nn0p1nn 12444	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 𝐾) ∈ ℕ₀ → ((𝑁 − 𝐾) + 1) ∈ ℕ)
93	24, 92	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝐾) + 1) ∈ ℕ)
94		elfzuz 13440	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁) → 𝑛 ∈ (ℤ_≥‘((𝑁 − 𝐾) + 1)))
95		eluznn 12835	. . . . . . . . . . . 12 ⊢ ((((𝑁 − 𝐾) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘((𝑁 − 𝐾) + 1))) → 𝑛 ∈ ℕ)
96	93, 94, 95	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → 𝑛 ∈ ℕ)
97	96, 43	syl 17	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘𝑛) ∈ ℂ)
98	91, 97	fsumcl 15660	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ)
99	50, 98	pncan2d 11498	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) − Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)) = Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛))
100	90, 99	eqtr2d 2773	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
101	100	fveq2d 6839	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) = (exp‘(Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))))
102	21	nnne0d 12199	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ≠ 0)
103		eflog 26545	. . . . . . . . 9 ⊢ (((!‘𝑁) ∈ ℂ ∧ (!‘𝑁) ≠ 0) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
104	22, 102, 103	syl2anc 585	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
105		logfac 26570	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → (log‘(!‘𝑁)) = Σ𝑛 ∈ (1...𝑁)(log‘𝑛))
106	20, 105	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘(!‘𝑁)) = Σ𝑛 ∈ (1...𝑁)(log‘𝑛))
107	106	fveq2d 6839	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
108	104, 107	eqtr3d 2774	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
109		eflog 26545	. . . . . . . . 9 ⊢ (((!‘(𝑁 − 𝐾)) ∈ ℂ ∧ (!‘(𝑁 − 𝐾)) ≠ 0) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (!‘(𝑁 − 𝐾)))
110	26, 27, 109	syl2anc 585	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (!‘(𝑁 − 𝐾)))
111		logfac 26570	. . . . . . . . . 10 ⊢ ((𝑁 − 𝐾) ∈ ℕ₀ → (log‘(!‘(𝑁 − 𝐾))) = Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))
112	24, 111	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘(!‘(𝑁 − 𝐾))) = Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))
113	112	fveq2d 6839	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
114	110, 113	eqtr3d 2774	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) = (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
115	108, 114	oveq12d 7378	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁 − 𝐾))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛))))
116	52, 101, 115	3eqtr4d 2782	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
117	32, 37, 116	3eqtr4d 2782	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)))
118	19, 117	eqtrd 2772	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)))
119		birthday.s	. . . . . . . 8 ⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}
120		mapvalg 8777	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑_m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)})
121	4, 3, 120	mp2an 693	. . . . . . . 8 ⊢ ((1...𝑁) ↑_m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}
122	119, 121	eqtr4i 2763	. . . . . . 7 ⊢ 𝑆 = ((1...𝑁) ↑_m (1...𝐾))
123	122	fveq2i 6838	. . . . . 6 ⊢ (♯‘𝑆) = (♯‘((1...𝑁) ↑_m (1...𝐾)))
124		hashmap 14362	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘((1...𝑁) ↑_m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾))))
125	4, 3, 124	mp2an 693	. . . . . 6 ⊢ (♯‘((1...𝑁) ↑_m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
126	123, 125	eqtri 2760	. . . . 5 ⊢ (♯‘𝑆) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
127	16, 11	oveq12d 7378	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))↑(♯‘(1...𝐾))) = (𝑁↑𝐾))
128	126, 127	eqtrid 2784	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (𝑁↑𝐾))
129		nncn 12157	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
130	129	adantr 480	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
131		nnne0 12183	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
132	131	adantr 480	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ≠ 0)
133		elfzelz 13444	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
134	133	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
135		explog 26563	. . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0 ∧ 𝐾 ∈ ℤ) → (𝑁↑𝐾) = (exp‘(𝐾 · (log‘𝑁))))
136	130, 132, 134, 135	syl3anc 1374	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁↑𝐾) = (exp‘(𝐾 · (log‘𝑁))))
137	128, 136	eqtrd 2772	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (exp‘(𝐾 · (log‘𝑁))))
138	118, 137	oveq12d 7378	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = ((exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
139	9	nn0cnd 12468	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℂ)
140		nnrp 12921	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
141	140	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ⁺)
142	141	relogcld 26592	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℝ)
143	142	recnd 11164	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℂ)
144	139, 143	mulcld 11156	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 · (log‘𝑁)) ∈ ℂ)
145		efsub 16029	. . 3 ⊢ ((Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ ∧ (𝐾 · (log‘𝑁)) ∈ ℂ) → (exp‘(Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁)))) = ((exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
146	98, 144, 145	syl2anc 585	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁)))) = ((exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
147		relogdiv 26562	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
148	41, 141, 147	syl2anr 598	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ ℕ) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
149	96, 148	syldan 592	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
150	149	sumeq2dv 15629	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)))
151	65	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
152	24	nn0zd 12517	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℤ)
153	152	peano2zd 12603	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝐾) + 1) ∈ ℤ)
154	96	nnrpd 12951	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → 𝑛 ∈ ℝ⁺)
155	141	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → 𝑁 ∈ ℝ⁺)
156	154, 155	rpdivcld 12970	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (𝑛 / 𝑁) ∈ ℝ⁺)
157	156	relogcld 26592	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℝ)
158	157	recnd 11164	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℂ)
159		fvoveq1 7383	. . . . . 6 ⊢ (𝑛 = (𝑁 − 𝑘) → (log‘(𝑛 / 𝑁)) = (log‘((𝑁 − 𝑘) / 𝑁)))
160	151, 153, 151, 158, 159	fsumrev 15706	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ ((𝑁 − 𝑁)...(𝑁 − ((𝑁 − 𝐾) + 1)))(log‘((𝑁 − 𝑘) / 𝑁)))
161	130	subidd 11484	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝑁) = 0)
162		1cnd 11131	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℂ)
163	130, 139, 162	subsubd 11524	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝐾 − 1)) = ((𝑁 − 𝐾) + 1))
164	163	oveq2d 7376	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝑁 − ((𝑁 − 𝐾) + 1)))
165		ax-1cn 11088	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
166		subcl 11383	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐾 − 1) ∈ ℂ)
167	139, 165, 166	sylancl 587	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 − 1) ∈ ℂ)
168	130, 167	nncand 11501	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝐾 − 1))
169	164, 168	eqtr3d 2774	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − ((𝑁 − 𝐾) + 1)) = (𝐾 − 1))
170	161, 169	oveq12d 7378	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝑁)...(𝑁 − ((𝑁 − 𝐾) + 1))) = (0...(𝐾 − 1)))
171	130	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
172		elfznn0 13540	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝐾 − 1)) → 𝑘 ∈ ℕ₀)
173	172	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℕ₀)
174	173	nn0cnd 12468	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℂ)
175	132	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ≠ 0)
176	171, 174, 171, 175	divsubdird 11960	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 − 𝑘) / 𝑁) = ((𝑁 / 𝑁) − (𝑘 / 𝑁)))
177	171, 175	dividd 11919	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (𝑁 / 𝑁) = 1)
178	177	oveq1d 7375	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 / 𝑁) − (𝑘 / 𝑁)) = (1 − (𝑘 / 𝑁)))
179	176, 178	eqtrd 2772	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 − 𝑘) / 𝑁) = (1 − (𝑘 / 𝑁)))
180	179	fveq2d 6839	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (log‘((𝑁 − 𝑘) / 𝑁)) = (log‘(1 − (𝑘 / 𝑁))))
181	170, 180	sumeq12rdv 15634	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑘 ∈ ((𝑁 − 𝑁)...(𝑁 − ((𝑁 − 𝐾) + 1)))(log‘((𝑁 − 𝑘) / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
182	160, 181	eqtrd 2772	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
183	143	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘𝑁) ∈ ℂ)
184	91, 97, 183	fsumsub 15715	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)) = (Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁)))
185		fsumconst 15717	. . . . . . . 8 ⊢ (((((𝑁 − 𝐾) + 1)...𝑁) ∈ Fin ∧ (log‘𝑁) ∈ ℂ) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁 − 𝐾) + 1)...𝑁)) · (log‘𝑁)))
186	91, 143, 185	syl2anc 585	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁 − 𝐾) + 1)...𝑁)) · (log‘𝑁)))
187		1zzd 12526	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℤ)
188		fzen 13461	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑁 − 𝐾) ∈ ℤ) → (1...𝐾) ≈ ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))))
189	187, 134, 152, 188	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))))
190	24	nn0cnd 12468	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℂ)
191		addcom 11323	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑁 − 𝐾) ∈ ℂ) → (1 + (𝑁 − 𝐾)) = ((𝑁 − 𝐾) + 1))
192	165, 190, 191	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1 + (𝑁 − 𝐾)) = ((𝑁 − 𝐾) + 1))
193	139, 130	pncan3d 11499	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 + (𝑁 − 𝐾)) = 𝑁)
194	192, 193	oveq12d 7378	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))) = (((𝑁 − 𝐾) + 1)...𝑁))
195	189, 194	breqtrd 5125	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ (((𝑁 − 𝐾) + 1)...𝑁))
196		hasheni 14275	. . . . . . . . . 10 ⊢ ((1...𝐾) ≈ (((𝑁 − 𝐾) + 1)...𝑁) → (♯‘(1...𝐾)) = (♯‘(((𝑁 − 𝐾) + 1)...𝑁)))
197	195, 196	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = (♯‘(((𝑁 − 𝐾) + 1)...𝑁)))
198	197, 11	eqtr3d 2774	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(((𝑁 − 𝐾) + 1)...𝑁)) = 𝐾)
199	198	oveq1d 7375	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(((𝑁 − 𝐾) + 1)...𝑁)) · (log‘𝑁)) = (𝐾 · (log‘𝑁)))
200	186, 199	eqtrd 2772	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁) = (𝐾 · (log‘𝑁)))
201	200	oveq2d 7376	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁)) = (Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))))
202	184, 201	eqtrd 2772	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)) = (Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))))
203	150, 182, 202	3eqtr3rd 2781	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
204	203	fveq2d 6839	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁)))) = (exp‘Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁)))))
205	138, 146, 204	3eqtr2d 2778	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = (exp‘Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∪ cun 3900 ∩ cin 3901 ∅c0 4286 class class class wbr 5099 ⟶wf 6489 –1-1→wf1 6490 ‘cfv 6493 (class class class)co 7360 ↑_m cmap 8767 ≈ cen 8884 Fincfn 8887 ℂcc 11028 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12149 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ℝ⁺crp 12909 ...cfz 13427 ↑cexp 13988 !cfa 14200 Ccbc 14229 ♯chash 14257 Σcsu 15613 expce 15988 logclog 26523

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-xnn0 12479 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-shft 14994 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-ef 15994 df-sin 15996 df-cos 15997 df-pi 15999 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-limc 25827 df-dv 25828 df-log 26525

This theorem is referenced by: birthdaylem3 26923

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