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Theorem birthdaylem2 26302
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 6845 . . . . . 6 (♯‘𝑇) = (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)})
3 fzfi 13877 . . . . . . 7 (1...𝐾) ∈ Fin
4 fzfi 13877 . . . . . . 7 (1...𝑁) ∈ Fin
5 hashf1 14356 . . . . . . 7 (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))))
63, 4, 5mp2an 690 . . . . . 6 (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
72, 6eqtri 2764 . . . . 5 (♯‘𝑇) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
8 elfznn0 13534 . . . . . . . . 9 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
98adantl 482 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ0)
10 hashfz1 14246 . . . . . . . 8 (𝐾 ∈ ℕ0 → (♯‘(1...𝐾)) = 𝐾)
119, 10syl 17 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = 𝐾)
1211fveq2d 6846 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(♯‘(1...𝐾))) = (!‘𝐾))
13 nnnn0 12420 . . . . . . . . 9 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
14 hashfz1 14246 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
1513, 14syl 17 . . . . . . . 8 (𝑁 ∈ ℕ → (♯‘(1...𝑁)) = 𝑁)
1615adantr 481 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝑁)) = 𝑁)
1716, 11oveq12d 7375 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))C(♯‘(1...𝐾))) = (𝑁C𝐾))
1812, 17oveq12d 7375 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))) = ((!‘𝐾) · (𝑁C𝐾)))
197, 18eqtrid 2788 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = ((!‘𝐾) · (𝑁C𝐾)))
2013adantr 481 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
2120faccld 14184 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
2221nncnd 12169 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
23 fznn0sub 13473 . . . . . . . . . 10 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ ℕ0)
2423adantl 482 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℕ0)
2524faccld 14184 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℕ)
2625nncnd 12169 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℂ)
2725nnne0d 12203 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ≠ 0)
2822, 26, 27divcld 11931 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁𝐾))) ∈ ℂ)
299faccld 14184 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
3029nncnd 12169 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
3129nnne0d 12203 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ≠ 0)
3228, 30, 31divcan2d 11933 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾))) = ((!‘𝑁) / (!‘(𝑁𝐾))))
33 bcval2 14205 . . . . . . . 8 (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3433adantl 482 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3522, 26, 30, 27, 31divdiv1d 11962 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3634, 35eqtr4d 2779 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾)))
3736oveq2d 7373 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾))))
38 fzfid 13878 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
39 elfznn 13470 . . . . . . . . . 10 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
4039adantl 482 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
41 nnrp 12926 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
4241relogcld 25978 . . . . . . . . . 10 (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℝ)
4342recnd 11183 . . . . . . . . 9 (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℂ)
4440, 43syl 17 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℂ)
4538, 44fsumcl 15618 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ)
46 fzfid 13878 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...(𝑁𝐾)) ∈ Fin)
47 elfznn 13470 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑁𝐾)) → 𝑛 ∈ ℕ)
4847adantl 482 . . . . . . . . 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 15618 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) ∈ ℂ)
51 efsub 15982 . . . . . . 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 584 . . . . . 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 12474 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℝ)
5453ltp1d 12085 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) < ((𝑁𝐾) + 1))
55 fzdisj 13468 . . . . . . . . . . 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 13548 . . . . . . . . . . . . . . . 16 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ (0...𝑁))
5857adantl 482 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ (0...𝑁))
59 elfzle2 13445 . . . . . . . . . . . . . . 15 ((𝑁𝐾) ∈ (0...𝑁) → (𝑁𝐾) ≤ 𝑁)
6058, 59syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ≤ 𝑁)
6160adantr 481 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ≤ 𝑁)
62 simpr 485 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ ℕ)
63 nnuz 12806 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6462, 63eleqtrdi 2848 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ (ℤ‘1))
65 nnz 12520 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
6665ad2antrr 724 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → 𝑁 ∈ ℤ)
67 elfz5 13433 . . . . . . . . . . . . . 14 (((𝑁𝐾) ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → ((𝑁𝐾) ∈ (1...𝑁) ↔ (𝑁𝐾) ≤ 𝑁))
6864, 66, 67syl2anc 584 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → ((𝑁𝐾) ∈ (1...𝑁) ↔ (𝑁𝐾) ≤ 𝑁))
6961, 68mpbird 256 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ (1...𝑁))
70 fzsplit 13467 . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (𝑁𝐾) = 0)
7372oveq2d 7373 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...(𝑁𝐾)) = (1...0))
74 fz10 13462 . . . . . . . . . . . . . 14 (1...0) = ∅
7573, 74eqtrdi 2792 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...(𝑁𝐾)) = ∅)
7675uneq1d 4122 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)) = (∅ ∪ (((𝑁𝐾) + 1)...𝑁)))
77 uncom 4113 . . . . . . . . . . . . . 14 (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = ((((𝑁𝐾) + 1)...𝑁) ∪ ∅)
78 un0 4350 . . . . . . . . . . . . . 14 ((((𝑁𝐾) + 1)...𝑁) ∪ ∅) = (((𝑁𝐾) + 1)...𝑁)
7977, 78eqtri 2764 . . . . . . . . . . . . 13 (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = (((𝑁𝐾) + 1)...𝑁)
8072oveq1d 7372 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((𝑁𝐾) + 1) = (0 + 1))
81 1e0p1 12660 . . . . . . . . . . . . . . 15 1 = (0 + 1)
8280, 81eqtr4di 2794 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((𝑁𝐾) + 1) = 1)
8382oveq1d 7372 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (((𝑁𝐾) + 1)...𝑁) = (1...𝑁))
8479, 83eqtrid 2788 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = (1...𝑁))
8576, 84eqtr2d 2777 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
86 elnn0 12415 . . . . . . . . . . . 12 ((𝑁𝐾) ∈ ℕ0 ↔ ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
8724, 86sylib 217 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
8871, 85, 87mpjaodan 957 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
8956, 88, 38, 44fsumsplit 15626 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
9089oveq1d 7372 . . . . . . . 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 13878 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁𝐾) + 1)...𝑁) ∈ Fin)
92 nn0p1nn 12452 . . . . . . . . . . . . 13 ((𝑁𝐾) ∈ ℕ0 → ((𝑁𝐾) + 1) ∈ ℕ)
9324, 92syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℕ)
94 elfzuz 13437 . . . . . . . . . . . 12 (𝑛 ∈ (((𝑁𝐾) + 1)...𝑁) → 𝑛 ∈ (ℤ‘((𝑁𝐾) + 1)))
95 eluznn 12843 . . . . . . . . . . . 12 ((((𝑁𝐾) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((𝑁𝐾) + 1))) → 𝑛 ∈ ℕ)
9693, 94, 95syl2an 596 . . . . . . . . . . 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 15618 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ)
9950, 98pncan2d 11514 . . . . . . . 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 2777 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
101100fveq2d 6846 . . . . . 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 12203 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ≠ 0)
103 eflog 25932 . . . . . . . . 9 (((!‘𝑁) ∈ ℂ ∧ (!‘𝑁) ≠ 0) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
10422, 102, 103syl2anc 584 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
105 logfac 25956 . . . . . . . . . 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 6846 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
108104, 107eqtr3d 2778 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
109 eflog 25932 . . . . . . . . 9 (((!‘(𝑁𝐾)) ∈ ℂ ∧ (!‘(𝑁𝐾)) ≠ 0) → (exp‘(log‘(!‘(𝑁𝐾)))) = (!‘(𝑁𝐾)))
11026, 27, 109syl2anc 584 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁𝐾)))) = (!‘(𝑁𝐾)))
111 logfac 25956 . . . . . . . . . 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 6846 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁𝐾)))) = (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
114110, 113eqtr3d 2778 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) = (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
115108, 114oveq12d 7375 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁𝐾))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))))
11652, 101, 1153eqtr4d 2786 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) = ((!‘𝑁) / (!‘(𝑁𝐾))))
11732, 37, 1163eqtr4d 2786 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
11819, 117eqtrd 2776 . . 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 8775 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑m (1...𝐾)) = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)})
1214, 3, 120mp2an 690 . . . . . . . 8 ((1...𝑁) ↑m (1...𝐾)) = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}
122119, 121eqtr4i 2767 . . . . . . 7 𝑆 = ((1...𝑁) ↑m (1...𝐾))
123122fveq2i 6845 . . . . . 6 (♯‘𝑆) = (♯‘((1...𝑁) ↑m (1...𝐾)))
124 hashmap 14335 . . . . . . 7 (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘((1...𝑁) ↑m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾))))
1254, 3, 124mp2an 690 . . . . . 6 (♯‘((1...𝑁) ↑m (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
126123, 125eqtri 2764 . . . . 5 (♯‘𝑆) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
12716, 11oveq12d 7375 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))↑(♯‘(1...𝐾))) = (𝑁𝐾))
128126, 127eqtrid 2788 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (𝑁𝐾))
129 nncn 12161 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
130129adantr 481 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
131 nnne0 12187 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
132131adantr 481 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ≠ 0)
133 elfzelz 13441 . . . . . 6 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
134133adantl 482 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
135 explog 25949 . . . . 5 ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0 ∧ 𝐾 ∈ ℤ) → (𝑁𝐾) = (exp‘(𝐾 · (log‘𝑁))))
136130, 132, 134, 135syl3anc 1371 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) = (exp‘(𝐾 · (log‘𝑁))))
137128, 136eqtrd 2776 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (exp‘(𝐾 · (log‘𝑁))))
138118, 137oveq12d 7375 . 2 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = ((exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
1399nn0cnd 12475 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℂ)
140 nnrp 12926 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
141140adantr 481 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ+)
142141relogcld 25978 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℝ)
143142recnd 11183 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℂ)
144139, 143mulcld 11175 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 · (log‘𝑁)) ∈ ℂ)
145 efsub 15982 . . 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 584 . 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 25948 . . . . . . 7 ((𝑛 ∈ ℝ+𝑁 ∈ ℝ+) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
14841, 141, 147syl2anr 597 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ ℕ) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
14996, 148syldan 591 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
150149sumeq2dv 15588 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)))
15165adantr 481 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
15224nn0zd 12525 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℤ)
153152peano2zd 12610 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℤ)
15496nnrpd 12955 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑛 ∈ ℝ+)
155141adantr 481 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑁 ∈ ℝ+)
156154, 155rpdivcld 12974 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (𝑛 / 𝑁) ∈ ℝ+)
157156relogcld 25978 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℝ)
158157recnd 11183 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℂ)
159 fvoveq1 7380 . . . . . 6 (𝑛 = (𝑁𝑘) → (log‘(𝑛 / 𝑁)) = (log‘((𝑁𝑘) / 𝑁)))
160151, 153, 151, 158, 159fsumrev 15664 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1)))(log‘((𝑁𝑘) / 𝑁)))
161130subidd 11500 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝑁) = 0)
162 1cnd 11150 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℂ)
163130, 139, 162subsubd 11540 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝐾 − 1)) = ((𝑁𝐾) + 1))
164163oveq2d 7373 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝑁 − ((𝑁𝐾) + 1)))
165 ax-1cn 11109 . . . . . . . . . 10 1 ∈ ℂ
166 subcl 11400 . . . . . . . . . 10 ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐾 − 1) ∈ ℂ)
167139, 165, 166sylancl 586 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 − 1) ∈ ℂ)
168130, 167nncand 11517 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝐾 − 1))
169164, 168eqtr3d 2778 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − ((𝑁𝐾) + 1)) = (𝐾 − 1))
170161, 169oveq12d 7375 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1))) = (0...(𝐾 − 1)))
171130adantr 481 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
172 elfznn0 13534 . . . . . . . . . . 11 (𝑘 ∈ (0...(𝐾 − 1)) → 𝑘 ∈ ℕ0)
173172adantl 482 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℕ0)
174173nn0cnd 12475 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℂ)
175132adantr 481 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ≠ 0)
176171, 174, 171, 175divsubdird 11970 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁𝑘) / 𝑁) = ((𝑁 / 𝑁) − (𝑘 / 𝑁)))
177171, 175dividd 11929 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (𝑁 / 𝑁) = 1)
178177oveq1d 7372 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 / 𝑁) − (𝑘 / 𝑁)) = (1 − (𝑘 / 𝑁)))
179176, 178eqtrd 2776 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁𝑘) / 𝑁) = (1 − (𝑘 / 𝑁)))
180179fveq2d 6846 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (log‘((𝑁𝑘) / 𝑁)) = (log‘(1 − (𝑘 / 𝑁))))
181170, 180sumeq12rdv 15592 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑘 ∈ ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1)))(log‘((𝑁𝑘) / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
182160, 181eqtrd 2776 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
183143adantr 481 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘𝑁) ∈ ℂ)
18491, 97, 183fsumsub 15673 . . . . 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 15675 . . . . . . . 8 (((((𝑁𝐾) + 1)...𝑁) ∈ Fin ∧ (log‘𝑁) ∈ ℂ) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)))
18691, 143, 185syl2anc 584 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)))
187 1zzd 12534 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℤ)
188 fzen 13458 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑁𝐾) ∈ ℤ) → (1...𝐾) ≈ ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))))
189187, 134, 152, 188syl3anc 1371 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))))
19024nn0cnd 12475 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℂ)
191 addcom 11341 . . . . . . . . . . . . 13 ((1 ∈ ℂ ∧ (𝑁𝐾) ∈ ℂ) → (1 + (𝑁𝐾)) = ((𝑁𝐾) + 1))
192165, 190, 191sylancr 587 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1 + (𝑁𝐾)) = ((𝑁𝐾) + 1))
193139, 130pncan3d 11515 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 + (𝑁𝐾)) = 𝑁)
194192, 193oveq12d 7375 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))) = (((𝑁𝐾) + 1)...𝑁))
195189, 194breqtrd 5131 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ (((𝑁𝐾) + 1)...𝑁))
196 hasheni 14248 . . . . . . . . . 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 2778 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(((𝑁𝐾) + 1)...𝑁)) = 𝐾)
199198oveq1d 7372 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)) = (𝐾 · (log‘𝑁)))
200186, 199eqtrd 2776 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = (𝐾 · (log‘𝑁)))
201200oveq2d 7373 . . . . 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 2776 . . . 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 2785 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
204203fveq2d 6846 . 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 2782 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 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2713  wne 2943  cun 3908  cin 3909  c0 4282   class class class wbr 5105  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  m cmap 8765  cen 8880  Fincfn 8883  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  0cn0 12413  cz 12499  cuz 12763  +crp 12915  ...cfz 13424  cexp 13967  !cfa 14173  Ccbc 14202  chash 14230  Σcsu 15570  expce 15944  logclog 25910
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912
This theorem is referenced by:  birthdaylem3  26303
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