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Theorem birthdaylem2 26862

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 6861	. . . . . 6 ⊢ (♯‘𝑇) = (♯‘{𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)})
3		fzfi 13937	. . . . . . 7 ⊢ (1...𝐾) ∈ Fin
4		fzfi 13937	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
5		hashf1 14422	. . . . . . 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 692	. . . . . 6 ⊢ (♯‘{𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
7	2, 6	eqtri 2752	. . . . 5 ⊢ (♯‘𝑇) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
8		elfznn0 13581	. . . . . . . . 9 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ₀)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ₀)
10		hashfz1 14311	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ₀ → (♯‘(1...𝐾)) = 𝐾)
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = 𝐾)
12	11	fveq2d 6862	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(♯‘(1...𝐾))) = (!‘𝐾))
13		nnnn0 12449	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 14311	. . . . . . . . 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 7405	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))C(♯‘(1...𝐾))) = (𝑁C𝐾))
18	12, 17	oveq12d 7405	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))) = ((!‘𝐾) · (𝑁C𝐾)))
19	7, 18	eqtrid 2776	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = ((!‘𝐾) · (𝑁C𝐾)))
20	13	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
21	20	faccld 14249	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
22	21	nncnd 12202	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
23		fznn0sub 13517	. . . . . . . . . 10 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ₀)
24	23	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℕ₀)
25	24	faccld 14249	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ∈ ℕ)
26	25	nncnd 12202	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ∈ ℂ)
27	25	nnne0d 12236	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) ≠ 0)
28	22, 26, 27	divcld 11958	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁 − 𝐾))) ∈ ℂ)
29	9	faccld 14249	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
30	29	nncnd 12202	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
31	29	nnne0d 12236	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ≠ 0)
32	28, 30, 31	divcan2d 11960	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾))) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
33		bcval2 14270	. . . . . . . 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 11989	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))
36	34, 35	eqtr4d 2767	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾)))
37	36	oveq2d 7403	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁 − 𝐾))) / (!‘𝐾))))
38		fzfid 13938	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
39		elfznn 13514	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
40	39	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
41		nnrp 12963	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
42	41	relogcld 26532	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℝ)
43	42	recnd 11202	. . . . . . . . 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 15699	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ)
46		fzfid 13938	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...(𝑁 − 𝐾)) ∈ Fin)
47		elfznn 13514	. . . . . . . . . 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 15699	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛) ∈ ℂ)
51		efsub 16068	. . . . . . 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 584	. . . . . 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 12504	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℝ)
54	53	ltp1d 12113	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) < ((𝑁 − 𝐾) + 1))
55		fzdisj 13512	. . . . . . . . . . 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 13596	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁))
58	57	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ (0...𝑁))
59		elfzle2 13489	. . . . . . . . . . . . . . 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 12836	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
64	62, 63	eleqtrdi 2838	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → (𝑁 − 𝐾) ∈ (ℤ_≥‘1))
65		nnz 12550	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
66	65	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) ∈ ℕ) → 𝑁 ∈ ℤ)
67		elfz5 13477	. . . . . . . . . . . . . 14 ⊢ (((𝑁 − 𝐾) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝐾) ∈ (1...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁))
68	64, 66, 67	syl2anc 584	. . . . . . . . . . . . 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 13511	. . . . . . . . . . . 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 7403	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...(𝑁 − 𝐾)) = (1...0))
74		fz10 13506	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
75	73, 74	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...(𝑁 − 𝐾)) = ∅)
76	75	uneq1d 4130	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
77		uncom 4121	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = ((((𝑁 − 𝐾) + 1)...𝑁) ∪ ∅)
78		un0 4357	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 − 𝐾) + 1)...𝑁) ∪ ∅) = (((𝑁 − 𝐾) + 1)...𝑁)
79	77, 78	eqtri 2752	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (((𝑁 − 𝐾) + 1)...𝑁)
80	72	oveq1d 7402	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((𝑁 − 𝐾) + 1) = (0 + 1))
81		1e0p1 12691	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
82	80, 81	eqtr4di 2782	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → ((𝑁 − 𝐾) + 1) = 1)
83	82	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (((𝑁 − 𝐾) + 1)...𝑁) = (1...𝑁))
84	79, 83	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (∅ ∪ (((𝑁 − 𝐾) + 1)...𝑁)) = (1...𝑁))
85	76, 84	eqtr2d 2765	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁 − 𝐾) = 0) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
86		elnn0 12444	. . . . . . . . . . . 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 960	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) = ((1...(𝑁 − 𝐾)) ∪ (((𝑁 − 𝐾) + 1)...𝑁)))
89	56, 88, 38, 44	fsumsplit 15707	. . . . . . . . 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 7402	. . . . . . . 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 13938	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁 − 𝐾) + 1)...𝑁) ∈ Fin)
92		nn0p1nn 12481	. . . . . . . . . . . . 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 13481	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁) → 𝑛 ∈ (ℤ_≥‘((𝑁 − 𝐾) + 1)))
95		eluznn 12877	. . . . . . . . . . . 12 ⊢ ((((𝑁 − 𝐾) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘((𝑁 − 𝐾) + 1))) → 𝑛 ∈ ℕ)
96	93, 94, 95	syl2an 596	. . . . . . . . . . 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 15699	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ)
99	50, 98	pncan2d 11535	. . . . . . . 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 2765	. . . . . . 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 6862	. . . . . 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 12236	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ≠ 0)
103		eflog 26485	. . . . . . . . 9 ⊢ (((!‘𝑁) ∈ ℂ ∧ (!‘𝑁) ≠ 0) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
104	22, 102, 103	syl2anc 584	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
105		logfac 26510	. . . . . . . . . 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 6862	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
108	104, 107	eqtr3d 2766	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
109		eflog 26485	. . . . . . . . 9 ⊢ (((!‘(𝑁 − 𝐾)) ∈ ℂ ∧ (!‘(𝑁 − 𝐾)) ≠ 0) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (!‘(𝑁 − 𝐾)))
110	26, 27, 109	syl2anc 584	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (!‘(𝑁 − 𝐾)))
111		logfac 26510	. . . . . . . . . 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 6862	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁 − 𝐾)))) = (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
114	110, 113	eqtr3d 2766	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁 − 𝐾)) = (exp‘Σ𝑛 ∈ (1...(𝑁 − 𝐾))(log‘𝑛)))
115	108, 114	oveq12d 7405	. . . . . 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 2774	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) = ((!‘𝑁) / (!‘(𝑁 − 𝐾))))
117	32, 37, 116	3eqtr4d 2774	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = (exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)))
118	19, 117	eqtrd 2764	. . 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 8809	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑_m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)})
121	4, 3, 120	mp2an 692	. . . . . . . 8 ⊢ ((1...𝑁) ↑_m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}
122	119, 121	eqtr4i 2755	. . . . . . 7 ⊢ 𝑆 = ((1...𝑁) ↑_m (1...𝐾))
123	122	fveq2i 6861	. . . . . 6 ⊢ (♯‘𝑆) = (♯‘((1...𝑁) ↑_m (1...𝐾)))
124		hashmap 14400	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘((1...𝑁) ↑_m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾))))
125	4, 3, 124	mp2an 692	. . . . . 6 ⊢ (♯‘((1...𝑁) ↑_m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
126	123, 125	eqtri 2752	. . . . 5 ⊢ (♯‘𝑆) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
127	16, 11	oveq12d 7405	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))↑(♯‘(1...𝐾))) = (𝑁↑𝐾))
128	126, 127	eqtrid 2776	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (𝑁↑𝐾))
129		nncn 12194	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
130	129	adantr 480	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
131		nnne0 12220	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
132	131	adantr 480	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ≠ 0)
133		elfzelz 13485	. . . . . 6 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
134	133	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
135		explog 26503	. . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0 ∧ 𝐾 ∈ ℤ) → (𝑁↑𝐾) = (exp‘(𝐾 · (log‘𝑁))))
136	130, 132, 134, 135	syl3anc 1373	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁↑𝐾) = (exp‘(𝐾 · (log‘𝑁))))
137	128, 136	eqtrd 2764	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (exp‘(𝐾 · (log‘𝑁))))
138	118, 137	oveq12d 7405	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = ((exp‘Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
139	9	nn0cnd 12505	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℂ)
140		nnrp 12963	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
141	140	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ⁺)
142	141	relogcld 26532	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℝ)
143	142	recnd 11202	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℂ)
144	139, 143	mulcld 11194	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 · (log‘𝑁)) ∈ ℂ)
145		efsub 16068	. . 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 584	. 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 26502	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
148	41, 141, 147	syl2anr 597	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ ℕ) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
149	96, 148	syldan 591	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
150	149	sumeq2dv 15668	. . . 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 12555	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℤ)
153	152	peano2zd 12641	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁 − 𝐾) + 1) ∈ ℤ)
154	96	nnrpd 12993	. . . . . . . . 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 13012	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (𝑛 / 𝑁) ∈ ℝ⁺)
157	156	relogcld 26532	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℝ)
158	157	recnd 11202	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℂ)
159		fvoveq1 7410	. . . . . 6 ⊢ (𝑛 = (𝑁 − 𝑘) → (log‘(𝑛 / 𝑁)) = (log‘((𝑁 − 𝑘) / 𝑁)))
160	151, 153, 151, 158, 159	fsumrev 15745	. . . . 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 11521	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝑁) = 0)
162		1cnd 11169	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℂ)
163	130, 139, 162	subsubd 11561	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝐾 − 1)) = ((𝑁 − 𝐾) + 1))
164	163	oveq2d 7403	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝑁 − ((𝑁 − 𝐾) + 1)))
165		ax-1cn 11126	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
166		subcl 11420	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐾 − 1) ∈ ℂ)
167	139, 165, 166	sylancl 586	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 − 1) ∈ ℂ)
168	130, 167	nncand 11538	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝐾 − 1))
169	164, 168	eqtr3d 2766	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − ((𝑁 − 𝐾) + 1)) = (𝐾 − 1))
170	161, 169	oveq12d 7405	. . . . . 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 13581	. . . . . . . . . . 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 12505	. . . . . . . . 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 11997	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 − 𝑘) / 𝑁) = ((𝑁 / 𝑁) − (𝑘 / 𝑁)))
177	171, 175	dividd 11956	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (𝑁 / 𝑁) = 1)
178	177	oveq1d 7402	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 / 𝑁) − (𝑘 / 𝑁)) = (1 − (𝑘 / 𝑁)))
179	176, 178	eqtrd 2764	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 − 𝑘) / 𝑁) = (1 − (𝑘 / 𝑁)))
180	179	fveq2d 6862	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (log‘((𝑁 − 𝑘) / 𝑁)) = (log‘(1 − (𝑘 / 𝑁))))
181	170, 180	sumeq12rdv 15673	. . . . 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 2764	. . . 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 15754	. . . . 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 15756	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁 − 𝐾) + 1)...𝑁)) · (log‘𝑁)))
187		1zzd 12564	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℤ)
188		fzen 13502	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑁 − 𝐾) ∈ ℤ) → (1...𝐾) ≈ ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))))
189	187, 134, 152, 188	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))))
190	24	nn0cnd 12505	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ ℂ)
191		addcom 11360	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑁 − 𝐾) ∈ ℂ) → (1 + (𝑁 − 𝐾)) = ((𝑁 − 𝐾) + 1))
192	165, 190, 191	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1 + (𝑁 − 𝐾)) = ((𝑁 − 𝐾) + 1))
193	139, 130	pncan3d 11536	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 + (𝑁 − 𝐾)) = 𝑁)
194	192, 193	oveq12d 7405	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1 + (𝑁 − 𝐾))...(𝐾 + (𝑁 − 𝐾))) = (((𝑁 − 𝐾) + 1)...𝑁))
195	189, 194	breqtrd 5133	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ (((𝑁 − 𝐾) + 1)...𝑁))
196		hasheni 14313	. . . . . . . . . 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 2766	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(((𝑁 − 𝐾) + 1)...𝑁)) = 𝐾)
199	198	oveq1d 7402	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(((𝑁 − 𝐾) + 1)...𝑁)) · (log‘𝑁)) = (𝐾 · (log‘𝑁)))
200	186, 199	eqtrd 2764	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁 − 𝐾) + 1)...𝑁)(log‘𝑁) = (𝐾 · (log‘𝑁)))
201	200	oveq2d 7403	. . . . 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 2764	. . . 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 2773	. . 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 6862	. 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 2770	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 847 = wceq 1540 ∈ wcel 2109 {cab 2707 ≠ wne 2925 ∪ cun 3912 ∩ cin 3913 ∅c0 4296 class class class wbr 5107 ⟶wf 6507 –1-1→wf1 6508 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 ≈ cen 8915 Fincfn 8918 ℂcc 11066 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 ...cfz 13468 ↑cexp 14026 !cfa 14238 Ccbc 14267 ♯chash 14295 Σcsu 15652 expce 16027 logclog 26463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465

This theorem is referenced by: birthdaylem3 26863

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