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Theorem birthdaylem2 25548
 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 = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}
birthday.t 𝑇 = {𝑓𝑓:(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.
StepHypRef Expression
1 birthday.t . . . . . . 7 𝑇 = {𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}
21fveq2i 6649 . . . . . 6 (♯‘𝑇) = (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)})
3 fzfi 13338 . . . . . . 7 (1...𝐾) ∈ Fin
4 fzfi 13338 . . . . . . 7 (1...𝑁) ∈ Fin
5 hashf1 13814 . . . . . . 7 (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))))
63, 4, 5mp2an 691 . . . . . 6 (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
72, 6eqtri 2821 . . . . 5 (♯‘𝑇) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
8 elfznn0 12998 . . . . . . . . 9 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
98adantl 485 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ0)
10 hashfz1 13705 . . . . . . . 8 (𝐾 ∈ ℕ0 → (♯‘(1...𝐾)) = 𝐾)
119, 10syl 17 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = 𝐾)
1211fveq2d 6650 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(♯‘(1...𝐾))) = (!‘𝐾))
13 nnnn0 11895 . . . . . . . . 9 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
14 hashfz1 13705 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
1513, 14syl 17 . . . . . . . 8 (𝑁 ∈ ℕ → (♯‘(1...𝑁)) = 𝑁)
1615adantr 484 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝑁)) = 𝑁)
1716, 11oveq12d 7154 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))C(♯‘(1...𝐾))) = (𝑁C𝐾))
1812, 17oveq12d 7154 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))) = ((!‘𝐾) · (𝑁C𝐾)))
197, 18syl5eq 2845 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = ((!‘𝐾) · (𝑁C𝐾)))
2013adantr 484 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
2120faccld 13643 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
2221nncnd 11644 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
23 fznn0sub 12937 . . . . . . . . . 10 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ ℕ0)
2423adantl 485 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℕ0)
2524faccld 13643 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℕ)
2625nncnd 11644 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℂ)
2725nnne0d 11678 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ≠ 0)
2822, 26, 27divcld 11408 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁𝐾))) ∈ ℂ)
299faccld 13643 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
3029nncnd 11644 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
3129nnne0d 11678 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ≠ 0)
3228, 30, 31divcan2d 11410 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾))) = ((!‘𝑁) / (!‘(𝑁𝐾))))
33 bcval2 13664 . . . . . . . 8 (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3433adantl 485 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3522, 26, 30, 27, 31divdiv1d 11439 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3634, 35eqtr4d 2836 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾)))
3736oveq2d 7152 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾))))
38 fzfid 13339 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
39 elfznn 12934 . . . . . . . . . 10 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
4039adantl 485 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
41 nnrp 12391 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
4241relogcld 25224 . . . . . . . . . 10 (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℝ)
4342recnd 10661 . . . . . . . . 9 (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℂ)
4440, 43syl 17 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℂ)
4538, 44fsumcl 15085 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ)
46 fzfid 13339 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...(𝑁𝐾)) ∈ Fin)
47 elfznn 12934 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑁𝐾)) → 𝑛 ∈ ℕ)
4847adantl 485 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...(𝑁𝐾))) → 𝑛 ∈ ℕ)
4948, 43syl 17 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...(𝑁𝐾))) → (log‘𝑛) ∈ ℂ)
5046, 49fsumcl 15085 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) ∈ ℂ)
51 efsub 15448 . . . . . . 7 ((Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ ∧ Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) ∈ ℂ) → (exp‘(Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))))
5245, 50, 51syl2anc 587 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))))
5324nn0red 11947 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℝ)
5453ltp1d 11562 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) < ((𝑁𝐾) + 1))
55 fzdisj 12932 . . . . . . . . . . 11 ((𝑁𝐾) < ((𝑁𝐾) + 1) → ((1...(𝑁𝐾)) ∩ (((𝑁𝐾) + 1)...𝑁)) = ∅)
5654, 55syl 17 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1...(𝑁𝐾)) ∩ (((𝑁𝐾) + 1)...𝑁)) = ∅)
57 fznn0sub2 13012 . . . . . . . . . . . . . . . 16 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ (0...𝑁))
5857adantl 485 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ (0...𝑁))
59 elfzle2 12909 . . . . . . . . . . . . . . 15 ((𝑁𝐾) ∈ (0...𝑁) → (𝑁𝐾) ≤ 𝑁)
6058, 59syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ≤ 𝑁)
6160adantr 484 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ≤ 𝑁)
62 simpr 488 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ ℕ)
63 nnuz 12272 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6462, 63eleqtrdi 2900 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ (ℤ‘1))
65 nnz 11995 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
6665ad2antrr 725 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → 𝑁 ∈ ℤ)
67 elfz5 12897 . . . . . . . . . . . . . 14 (((𝑁𝐾) ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → ((𝑁𝐾) ∈ (1...𝑁) ↔ (𝑁𝐾) ≤ 𝑁))
6864, 66, 67syl2anc 587 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → ((𝑁𝐾) ∈ (1...𝑁) ↔ (𝑁𝐾) ≤ 𝑁))
6961, 68mpbird 260 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ (1...𝑁))
70 fzsplit 12931 . . . . . . . . . . . 12 ((𝑁𝐾) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
7169, 70syl 17 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
72 simpr 488 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (𝑁𝐾) = 0)
7372oveq2d 7152 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...(𝑁𝐾)) = (1...0))
74 fz10 12926 . . . . . . . . . . . . . 14 (1...0) = ∅
7573, 74eqtrdi 2849 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...(𝑁𝐾)) = ∅)
7675uneq1d 4089 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)) = (∅ ∪ (((𝑁𝐾) + 1)...𝑁)))
77 uncom 4080 . . . . . . . . . . . . . 14 (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = ((((𝑁𝐾) + 1)...𝑁) ∪ ∅)
78 un0 4298 . . . . . . . . . . . . . 14 ((((𝑁𝐾) + 1)...𝑁) ∪ ∅) = (((𝑁𝐾) + 1)...𝑁)
7977, 78eqtri 2821 . . . . . . . . . . . . 13 (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = (((𝑁𝐾) + 1)...𝑁)
8072oveq1d 7151 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((𝑁𝐾) + 1) = (0 + 1))
81 1e0p1 12131 . . . . . . . . . . . . . . 15 1 = (0 + 1)
8280, 81eqtr4di 2851 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((𝑁𝐾) + 1) = 1)
8382oveq1d 7151 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (((𝑁𝐾) + 1)...𝑁) = (1...𝑁))
8479, 83syl5eq 2845 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = (1...𝑁))
8576, 84eqtr2d 2834 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
86 elnn0 11890 . . . . . . . . . . . 12 ((𝑁𝐾) ∈ ℕ0 ↔ ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
8724, 86sylib 221 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
8871, 85, 87mpjaodan 956 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
8956, 88, 38, 44fsumsplit 15092 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
9089oveq1d 7151 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)) = ((Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
91 fzfid 13339 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁𝐾) + 1)...𝑁) ∈ Fin)
92 nn0p1nn 11927 . . . . . . . . . . . . 13 ((𝑁𝐾) ∈ ℕ0 → ((𝑁𝐾) + 1) ∈ ℕ)
9324, 92syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℕ)
94 elfzuz 12901 . . . . . . . . . . . 12 (𝑛 ∈ (((𝑁𝐾) + 1)...𝑁) → 𝑛 ∈ (ℤ‘((𝑁𝐾) + 1)))
95 eluznn 12309 . . . . . . . . . . . 12 ((((𝑁𝐾) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((𝑁𝐾) + 1))) → 𝑛 ∈ ℕ)
9693, 94, 95syl2an 598 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑛 ∈ ℕ)
9796, 43syl 17 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘𝑛) ∈ ℂ)
9891, 97fsumcl 15085 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ)
9950, 98pncan2d 10991 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)) = Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛))
10090, 99eqtr2d 2834 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
101100fveq2d 6650 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) = (exp‘(Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))))
10221nnne0d 11678 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ≠ 0)
103 eflog 25178 . . . . . . . . 9 (((!‘𝑁) ∈ ℂ ∧ (!‘𝑁) ≠ 0) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
10422, 102, 103syl2anc 587 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
105 logfac 25202 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑛 ∈ (1...𝑁)(log‘𝑛))
10620, 105syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘(!‘𝑁)) = Σ𝑛 ∈ (1...𝑁)(log‘𝑛))
107106fveq2d 6650 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
108104, 107eqtr3d 2835 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
109 eflog 25178 . . . . . . . . 9 (((!‘(𝑁𝐾)) ∈ ℂ ∧ (!‘(𝑁𝐾)) ≠ 0) → (exp‘(log‘(!‘(𝑁𝐾)))) = (!‘(𝑁𝐾)))
11026, 27, 109syl2anc 587 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁𝐾)))) = (!‘(𝑁𝐾)))
111 logfac 25202 . . . . . . . . . 10 ((𝑁𝐾) ∈ ℕ0 → (log‘(!‘(𝑁𝐾))) = Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))
11224, 111syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘(!‘(𝑁𝐾))) = Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))
113112fveq2d 6650 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁𝐾)))) = (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
114110, 113eqtr3d 2835 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) = (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
115108, 114oveq12d 7154 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁𝐾))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))))
11652, 101, 1153eqtr4d 2843 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) = ((!‘𝑁) / (!‘(𝑁𝐾))))
11732, 37, 1163eqtr4d 2843 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
11819, 117eqtrd 2833 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
119 birthday.s . . . . . . . 8 𝑆 = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}
120 mapvalg 8402 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑m (1...𝐾)) = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)})
1214, 3, 120mp2an 691 . . . . . . . 8 ((1...𝑁) ↑m (1...𝐾)) = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}
122119, 121eqtr4i 2824 . . . . . . 7 𝑆 = ((1...𝑁) ↑m (1...𝐾))
123122fveq2i 6649 . . . . . 6 (♯‘𝑆) = (♯‘((1...𝑁) ↑m (1...𝐾)))
124 hashmap 13795 . . . . . . 7 (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘((1...𝑁) ↑m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾))))
1254, 3, 124mp2an 691 . . . . . 6 (♯‘((1...𝑁) ↑m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
126123, 125eqtri 2821 . . . . 5 (♯‘𝑆) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
12716, 11oveq12d 7154 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))↑(♯‘(1...𝐾))) = (𝑁𝐾))
128126, 127syl5eq 2845 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (𝑁𝐾))
129 nncn 11636 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
130129adantr 484 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
131 nnne0 11662 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
132131adantr 484 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ≠ 0)
133 elfzelz 12905 . . . . . 6 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
134133adantl 485 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
135 explog 25195 . . . . 5 ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0 ∧ 𝐾 ∈ ℤ) → (𝑁𝐾) = (exp‘(𝐾 · (log‘𝑁))))
136130, 132, 134, 135syl3anc 1368 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) = (exp‘(𝐾 · (log‘𝑁))))
137128, 136eqtrd 2833 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (exp‘(𝐾 · (log‘𝑁))))
138118, 137oveq12d 7154 . 2 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = ((exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
1399nn0cnd 11948 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℂ)
140 nnrp 12391 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
141140adantr 484 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ+)
142141relogcld 25224 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℝ)
143142recnd 10661 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℂ)
144139, 143mulcld 10653 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 · (log‘𝑁)) ∈ ℂ)
145 efsub 15448 . . 3 ((Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ ∧ (𝐾 · (log‘𝑁)) ∈ ℂ) → (exp‘(Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁)))) = ((exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
14698, 144, 145syl2anc 587 . 2 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁)))) = ((exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
147 relogdiv 25194 . . . . . . 7 ((𝑛 ∈ ℝ+𝑁 ∈ ℝ+) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
14841, 141, 147syl2anr 599 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ ℕ) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
14996, 148syldan 594 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
150149sumeq2dv 15055 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)))
15165adantr 484 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
15224nn0zd 12076 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℤ)
153152peano2zd 12081 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℤ)
15496nnrpd 12420 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑛 ∈ ℝ+)
155141adantr 484 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑁 ∈ ℝ+)
156154, 155rpdivcld 12439 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (𝑛 / 𝑁) ∈ ℝ+)
157156relogcld 25224 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℝ)
158157recnd 10661 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℂ)
159 fvoveq1 7159 . . . . . 6 (𝑛 = (𝑁𝑘) → (log‘(𝑛 / 𝑁)) = (log‘((𝑁𝑘) / 𝑁)))
160151, 153, 151, 158, 159fsumrev 15129 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1)))(log‘((𝑁𝑘) / 𝑁)))
161130subidd 10977 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝑁) = 0)
162 1cnd 10628 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℂ)
163130, 139, 162subsubd 11017 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝐾 − 1)) = ((𝑁𝐾) + 1))
164163oveq2d 7152 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝑁 − ((𝑁𝐾) + 1)))
165 ax-1cn 10587 . . . . . . . . . 10 1 ∈ ℂ
166 subcl 10877 . . . . . . . . . 10 ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐾 − 1) ∈ ℂ)
167139, 165, 166sylancl 589 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 − 1) ∈ ℂ)
168130, 167nncand 10994 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝐾 − 1))
169164, 168eqtr3d 2835 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − ((𝑁𝐾) + 1)) = (𝐾 − 1))
170161, 169oveq12d 7154 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1))) = (0...(𝐾 − 1)))
171130adantr 484 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
172 elfznn0 12998 . . . . . . . . . . 11 (𝑘 ∈ (0...(𝐾 − 1)) → 𝑘 ∈ ℕ0)
173172adantl 485 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℕ0)
174173nn0cnd 11948 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℂ)
175132adantr 484 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ≠ 0)
176171, 174, 171, 175divsubdird 11447 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁𝑘) / 𝑁) = ((𝑁 / 𝑁) − (𝑘 / 𝑁)))
177171, 175dividd 11406 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (𝑁 / 𝑁) = 1)
178177oveq1d 7151 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 / 𝑁) − (𝑘 / 𝑁)) = (1 − (𝑘 / 𝑁)))
179176, 178eqtrd 2833 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁𝑘) / 𝑁) = (1 − (𝑘 / 𝑁)))
180179fveq2d 6650 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (log‘((𝑁𝑘) / 𝑁)) = (log‘(1 − (𝑘 / 𝑁))))
181170, 180sumeq12rdv 15059 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑘 ∈ ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1)))(log‘((𝑁𝑘) / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
182160, 181eqtrd 2833 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
183143adantr 484 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘𝑁) ∈ ℂ)
18491, 97, 183fsumsub 15138 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)) = (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁)))
185 fsumconst 15140 . . . . . . . 8 (((((𝑁𝐾) + 1)...𝑁) ∈ Fin ∧ (log‘𝑁) ∈ ℂ) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)))
18691, 143, 185syl2anc 587 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)))
187 1zzd 12004 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℤ)
188 fzen 12922 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑁𝐾) ∈ ℤ) → (1...𝐾) ≈ ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))))
189187, 134, 152, 188syl3anc 1368 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))))
19024nn0cnd 11948 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℂ)
191 addcom 10818 . . . . . . . . . . . . 13 ((1 ∈ ℂ ∧ (𝑁𝐾) ∈ ℂ) → (1 + (𝑁𝐾)) = ((𝑁𝐾) + 1))
192165, 190, 191sylancr 590 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1 + (𝑁𝐾)) = ((𝑁𝐾) + 1))
193139, 130pncan3d 10992 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 + (𝑁𝐾)) = 𝑁)
194192, 193oveq12d 7154 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))) = (((𝑁𝐾) + 1)...𝑁))
195189, 194breqtrd 5057 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ (((𝑁𝐾) + 1)...𝑁))
196 hasheni 13707 . . . . . . . . . 10 ((1...𝐾) ≈ (((𝑁𝐾) + 1)...𝑁) → (♯‘(1...𝐾)) = (♯‘(((𝑁𝐾) + 1)...𝑁)))
197195, 196syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = (♯‘(((𝑁𝐾) + 1)...𝑁)))
198197, 11eqtr3d 2835 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(((𝑁𝐾) + 1)...𝑁)) = 𝐾)
199198oveq1d 7151 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)) = (𝐾 · (log‘𝑁)))
200186, 199eqtrd 2833 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = (𝐾 · (log‘𝑁)))
201200oveq2d 7152 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁)) = (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))))
202184, 201eqtrd 2833 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)) = (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))))
203150, 182, 2023eqtr3rd 2842 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
204203fveq2d 6650 . 2 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁)))) = (exp‘Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁)))))
205138, 146, 2043eqtr2d 2839 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 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {cab 2776   ≠ wne 2987   ∪ cun 3879   ∩ cin 3880  ∅c0 4243   class class class wbr 5031  ⟶wf 6321  –1-1→wf1 6322  ‘cfv 6325  (class class class)co 7136   ↑m cmap 8392   ≈ cen 8492  Fincfn 8495  ℂcc 10527  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534   < clt 10667   ≤ cle 10668   − cmin 10862   / cdiv 11289  ℕcn 11628  ℕ0cn0 11888  ℤcz 11972  ℤ≥cuz 12234  ℝ+crp 12380  ...cfz 12888  ↑cexp 13428  !cfa 13632  Ccbc 13661  ♯chash 13689  Σcsu 15037  expce 15410  logclog 25156 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-of 7391  df-om 7564  df-1st 7674  df-2nd 7675  df-supp 7817  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-er 8275  df-map 8394  df-pm 8395  df-ixp 8448  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fsupp 8821  df-fi 8862  df-sup 8893  df-inf 8894  df-oi 8961  df-dju 9317  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-xnn0 11959  df-z 11973  df-dec 12090  df-uz 12235  df-q 12340  df-rp 12381  df-xneg 12498  df-xadd 12499  df-xmul 12500  df-ioo 12733  df-ioc 12734  df-ico 12735  df-icc 12736  df-fz 12889  df-fzo 13032  df-fl 13160  df-mod 13236  df-seq 13368  df-exp 13429  df-fac 13633  df-bc 13662  df-hash 13690  df-shft 14421  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-limsup 14823  df-clim 14840  df-rlim 14841  df-sum 15038  df-ef 15416  df-sin 15418  df-cos 15419  df-pi 15421  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-hom 16584  df-cco 16585  df-rest 16691  df-topn 16692  df-0g 16710  df-gsum 16711  df-topgen 16712  df-pt 16713  df-prds 16716  df-xrs 16770  df-qtop 16775  df-imas 16776  df-xps 16778  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-mulg 18221  df-cntz 18443  df-cmn 18904  df-psmet 20087  df-xmet 20088  df-met 20089  df-bl 20090  df-mopn 20091  df-fbas 20092  df-fg 20093  df-cnfld 20096  df-top 21509  df-topon 21526  df-topsp 21548  df-bases 21561  df-cld 21634  df-ntr 21635  df-cls 21636  df-nei 21713  df-lp 21751  df-perf 21752  df-cn 21842  df-cnp 21843  df-haus 21930  df-tx 22177  df-hmeo 22370  df-fil 22461  df-fm 22553  df-flim 22554  df-flf 22555  df-xms 22937  df-ms 22938  df-tms 22939  df-cncf 23493  df-limc 24479  df-dv 24480  df-log 25158 This theorem is referenced by:  birthdaylem3  25549
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