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Theorem birthdaylem2 27019

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 6872	. . . . . 6 ⊢ (♯‘𝑇) = (♯‘{𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)})
3		fzfi 13987	. . . . . . 7 ⊢ (1...𝐾) ∈ Fin
4		fzfi 13987	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
5		hashf1 14472	. . . . . . 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 702	. . . . . 6 ⊢ (♯‘{𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
7	2, 6	eqtri 2787	. . . . 5 ⊢ (♯‘𝑇) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
8		elfznn0 13627	. . . . . . . . 9 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
9	8	adantl 485	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ₀)
10		hashfz1 14361	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ₀ → (♯‘(1...𝐾)) = 𝐾)
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = 𝐾)
12	11	fveq2d 6873	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(♯‘(1...𝐾))) = (!‘𝐾))
13		nnnn0 12490	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 14361	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (♯‘(1...𝑁)) = 𝑁)
16	15	adantr 484	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝑁)) = 𝑁)
17	16, 11	oveq12d 7416	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))C(♯‘(1...𝐾))) = (𝑁C𝐾))
18	12, 17	oveq12d 7416	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))) = ((!‘𝐾) · (𝑁C𝐾)))
19	7, 18	eqtrid 2811	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = ((!‘𝐾) · (𝑁C𝐾)))
20	13	adantr 484	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
21	20	faccld 14299	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
22	21	nncnd 12228	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
23		fznn0sub 13563	. . . . . . . . . 10 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
24	23	adantl 485	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℕ₀)
25	24	faccld 14299	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
26	25	nncnd 12228	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
27	25	nnne0d 12265	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ≠ 0)
28	22, 26, 27	divcld 11969	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁 − 𝐾))) ∈ ℂ)
29	9	faccld 14299	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
30	29	nncnd 12228	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
31	29	nnne0d 12265	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ≠ 0)
32	28, 30, 31	divcan2d 11971	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾))) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
33		bcval2 14320	. . . . . . . 8 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
34	33	adantl 485	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
35	22, 26, 30, 27, 31	divdiv1d 12000	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
36	34, 35	eqtr4d 2802	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
37	36	oveq2d 7414	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾))))
38		fzfid 13988	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
39		elfznn 13560	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
40	39	adantl 485	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
41		nnrp 13007	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
42	41	relogcld 26690	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℝ)
43	42	recnd 11212	. . . . . . . . 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 15762	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ)
46		fzfid 13988	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...(𝑁 − 𝐾)) ∈ Fin)
47		elfznn 13560	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(𝑁 − 𝐾)) → 𝑛 ∈ ℕ)
48	47	adantl 485	. . . . . . . . 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 15762	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛) ∈ ℂ)
51		efsub 16134	. . . . . . 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 593	. . . . . 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 12545	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℝ)
54	53	ltp1d 12124	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) < ((𝑁 − 𝐾) + 1))
55		fzdisj 13558	. . . . . . . . . . 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 13642	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁))
58	57	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ (0...𝑁))
59		elfzle2 13535	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ≤ 𝑁)
62		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ∈ ℕ)
63		nnuz 12880	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
64	62, 63	eleqtrdi 2874	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ∈ (ℤ_≥‘1))
65		nnz 12591	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
66	65	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → 𝑁 ∈ ℤ)
67		elfz5 13523	. . . . . . . . . . . . . 14 ⊢ (((𝑁 − 𝐾) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝐾) ∈ (1...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁))
68	64, 66, 67	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → ((𝑁 − 𝐾) ∈ (1...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁))
69	61, 68	mpbird 259	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ∈ (1...𝑁))
70		fzsplit 13557	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (𝑁 − 𝐾) = 0)
73	72	oveq2d 7414	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...(𝑁 − 𝐾)) = (1...0))
74		fz10 13552	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
75	73, 74	eqtrdi 2815	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...(𝑁 − 𝐾)) = ∅)
76	75	uneq1d 4122	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
77		uncom 4113	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = ((((𝑁 − 𝐾) + 1)...𝑁) ∪ ∅)
78		un0 4350	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 − 𝐾) + 1)...𝑁) ∪ ∅) = (((𝑁 − 𝐾) + 1)...𝑁)
79	77, 78	eqtri 2787	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (((𝑁 − 𝐾) + 1)...𝑁)
80	72	oveq1d 7413	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((𝑁 − 𝐾) + 1) = (0 + 1))
81		1e0p1 12737	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
82	80, 81	eqtr4di 2817	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((𝑁 − 𝐾) + 1) = 1)
83	82	oveq1d 7413	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (((𝑁 − 𝐾) + 1)...𝑁) = (1...𝑁))
84	79, 83	eqtrid 2811	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (1...𝑁))
85	76, 84	eqtr2d 2800	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
86		elnn0 12485	. . . . . . . . . . . 12 ⊢ ((𝑁 − 𝐾) ∈ ℕ₀ ↔ ((𝑁 − 𝐾) ∈ ℕ ∨ (𝑁 − 𝐾) = 0))
87	24, 86	sylib 220	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝐾) ∈ ℕ ∨ (𝑁 − 𝐾) = 0))
88	71, 85, 87	mpjaodan 971	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
89	56, 88, 38, 44	fsumsplit 15770	. . . . . . . . 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 7413	. . . . . . . 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 13988	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁 − 𝐾) + 1)...𝑁) ∈ Fin)
92		nn0p1nn 12522	. . . . . . . . . . . . 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 13527	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁) → 𝑛 ∈ (ℤ_≥‘((𝑁 − 𝐾) + 1)))
95		eluznn 12921	. . . . . . . . . . . 12 ⊢ ((((𝑁 − 𝐾) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘((𝑁 − 𝐾) + 1))) → 𝑛 ∈ ℕ)
96	93, 94, 95	syl2an 605	. . . . . . . . . . 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 15762	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ)
99	50, 98	pncan2d 11546	. . . . . . . 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 2800	. . . . . . 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 6873	. . . . . 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 12265	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ≠ 0)
103		eflog 26643	. . . . . . . . 9 ⊢ (((!‘𝑁) ∈ ℂ ∧ (!‘𝑁) ≠ 0) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
104	22, 102, 103	syl2anc 593	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
105		logfac 26668	. . . . . . . . . 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 6873	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
108	104, 107	eqtr3d 2801	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
109		eflog 26643	. . . . . . . . 9 ⊢ (((!‘(𝑁 − 𝐾)) ∈ ℂ ∧ (!‘(𝑁 − 𝐾)) ≠ 0) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (!‘(𝑁 − 𝐾)))
110	26, 27, 109	syl2anc 593	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (!‘(𝑁 − 𝐾)))
111		logfac 26668	. . . . . . . . . 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 6873	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
114	110, 113	eqtr3d 2801	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) = (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
115	108, 114	oveq12d 7416	. . . . . 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 2809	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
117	32, 37, 116	3eqtr4d 2809	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)))
118	19, 117	eqtrd 2799	. . 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 8819	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑_m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)})
121	4, 3, 120	mp2an 702	. . . . . . . 8 ⊢ ((1...𝑁) ↑_m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}
122	119, 121	eqtr4i 2790	. . . . . . 7 ⊢ 𝑆 = ((1...𝑁) ↑_m (1...𝐾))
123	122	fveq2i 6872	. . . . . 6 ⊢ (♯‘𝑆) = (♯‘((1...𝑁) ↑_m (1...𝐾)))
124		hashmap 14450	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘((1...𝑁) ↑_m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾))))
125	4, 3, 124	mp2an 702	. . . . . 6 ⊢ (♯‘((1...𝑁) ↑_m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
126	123, 125	eqtri 2787	. . . . 5 ⊢ (♯‘𝑆) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
127	16, 11	oveq12d 7416	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))↑(♯‘(1...𝐾))) = (𝑁↑𝐾))
128	126, 127	eqtrid 2811	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (𝑁↑𝐾))
129		nncn 12220	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
130	129	adantr 484	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
131		nnne0 12249	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
132	131	adantr 484	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ≠ 0)
133		elfzelz 13531	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
134	133	adantl 485	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
135		explog 26661	. . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0 ∧ 𝐾 ∈ ℤ) → (𝑁↑𝐾) = (exp‘(𝐾 · (log‘𝑁))))
136	130, 132, 134, 135	syl3anc 1392	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁↑𝐾) = (exp‘(𝐾 · (log‘𝑁))))
137	128, 136	eqtrd 2799	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (exp‘(𝐾 · (log‘𝑁))))
138	118, 137	oveq12d 7416	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = ((exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
139	9	nn0cnd 12546	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℂ)
140		nnrp 13007	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
141	140	adantr 484	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ⁺)
142	141	relogcld 26690	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℝ)
143	142	recnd 11212	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℂ)
144	139, 143	mulcld 11204	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 · (log‘𝑁)) ∈ ℂ)
145		efsub 16134	. . 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 593	. 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 26660	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
148	41, 141, 147	syl2anr 606	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ ℕ) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
149	96, 148	syldan 600	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
150	149	sumeq2dv 15731	. . . 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 484	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
152	24	nn0zd 12595	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℤ)
153	152	peano2zd 12682	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝐾) + 1) ∈ ℤ)
154	96	nnrpd 13037	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → 𝑛 ∈ ℝ⁺)
155	141	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → 𝑁 ∈ ℝ⁺)
156	154, 155	rpdivcld 13056	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (𝑛 / 𝑁) ∈ ℝ⁺)
157	156	relogcld 26690	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℝ)
158	157	recnd 11212	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℂ)
159		fvoveq1 7421	. . . . . 6 ⊢ (𝑛 = (𝑁 − 𝑘) → (log‘(𝑛 / 𝑁)) = (log‘((𝑁 − 𝑘) / 𝑁)))
160	151, 153, 151, 158, 159	fsumrev 15808	. . . . 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 11532	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝑁) = 0)
162		1cnd 11177	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℂ)
163	130, 139, 162	subsubd 11572	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝐾 − 1)) = ((𝑁 − 𝐾) + 1))
164	163	oveq2d 7414	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝑁 − ((𝑁 − 𝐾) + 1)))
165		ax-1cn 11133	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
166		subcl 11431	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐾 − 1) ∈ ℂ)
167	139, 165, 166	sylancl 595	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 − 1) ∈ ℂ)
168	130, 167	nncand 11549	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝐾 − 1))
169	164, 168	eqtr3d 2801	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − ((𝑁 − 𝐾) + 1)) = (𝐾 − 1))
170	161, 169	oveq12d 7416	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝑁)...(𝑁 − ((𝑁 − 𝐾) + 1))) = (0...(𝐾 − 1)))
171	130	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
172		elfznn0 13627	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...(𝐾 − 1)) → 𝑘 ∈ ℕ₀)
173	172	adantl 485	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℕ₀)
174	173	nn0cnd 12546	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℂ)
175	132	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ≠ 0)
176	171, 174, 171, 175	divsubdird 12008	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 − 𝑘) / 𝑁) = ((𝑁 / 𝑁) − (𝑘 / 𝑁)))
177	171, 175	dividd 11967	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (𝑁 / 𝑁) = 1)
178	177	oveq1d 7413	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 / 𝑁) − (𝑘 / 𝑁)) = (1 − (𝑘 / 𝑁)))
179	176, 178	eqtrd 2799	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 − 𝑘) / 𝑁) = (1 − (𝑘 / 𝑁)))
180	179	fveq2d 6873	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (log‘((𝑁 − 𝑘) / 𝑁)) = (log‘(1 − (𝑘 / 𝑁))))
181	170, 180	sumeq12rdv 15736	. . . . 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 2799	. . . 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 484	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘𝑁) ∈ ℂ)
184	91, 97, 183	fsumsub 15817	. . . . 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 15819	. . . . . . . 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 593	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁 − 𝐾) + 1)...𝑁)) · (log‘𝑁)))
187		1zzd 12604	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℤ)
188		fzen 13548	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑁 − 𝐾) ∈ ℤ) → (1...𝐾) ≈ ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))))
189	187, 134, 152, 188	syl3anc 1392	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))))
190	24	nn0cnd 12546	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℂ)
191		addcom 11371	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑁 − 𝐾) ∈ ℂ) → (1 + (𝑁 − 𝐾)) = ((𝑁 − 𝐾) + 1))
192	165, 190, 191	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1 + (𝑁 − 𝐾)) = ((𝑁 − 𝐾) + 1))
193	139, 130	pncan3d 11547	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 + (𝑁 − 𝐾)) = 𝑁)
194	192, 193	oveq12d 7416	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))) = (((𝑁 − 𝐾) + 1)...𝑁))
195	189, 194	breqtrd 5128	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ (((𝑁 − 𝐾) + 1)...𝑁))
196		hasheni 14363	. . . . . . . . . 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 2801	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(((𝑁 − 𝐾) + 1)...𝑁)) = 𝐾)
199	198	oveq1d 7413	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(((𝑁 − 𝐾) + 1)...𝑁)) · (log‘𝑁)) = (𝐾 · (log‘𝑁)))
200	186, 199	eqtrd 2799	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁) = (𝐾 · (log‘𝑁)))
201	200	oveq2d 7414	. . . . 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 2799	. . . 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 2808	. . 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 6873	. 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 2805	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 208 ∧ wa 399 ∨ wo 858 = wceq 1562 ∈ wcel 2144 {cab 2742 ≠ wne 2959 ∪ cun 3904 ∩ cin 3905 ∅c0 4287 class class class wbr 5102 ⟶wf 6519 –1-1→wf1 6520 ‘cfv 6523 (class class class)co 7398 ↑_m cmap 8810 ≈ cen 8926 Fincfn 8929 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11218 ≤ cle 11219 − cmin 11416 / cdiv 11846 ℕcn 12212 ℕ₀cn0 12483 ℤcz 12570 ℤ_≥cuz 12841 ℝ⁺crp 12995 ...cfz 13514 ↑cexp 14076 !cfa 14288 Ccbc 14317 ♯chash 14345 Σcsu 15715 expce 16093 logclog 26621

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-oadd 8443 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9861 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-xnn0 12557 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-fl 13804 df-mod 13882 df-seq 14017 df-exp 14077 df-fac 14289 df-bc 14318 df-hash 14346 df-shft 15082 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-limsup 15500 df-clim 15517 df-rlim 15518 df-sum 15716 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cn 23289 df-cnp 23290 df-haus 23377 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-xms 24382 df-ms 24383 df-tms 24384 df-cncf 24942 df-limc 25930 df-dv 25931 df-log 26623

This theorem is referenced by: birthdaylem3 27020

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