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Theorem birthdaylem2 24970
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 6378 . . . . . 6 (♯‘𝑇) = (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)})
3 fzfi 12979 . . . . . . 7 (1...𝐾) ∈ Fin
4 fzfi 12979 . . . . . . 7 (1...𝑁) ∈ Fin
5 hashf1 13442 . . . . . . 7 (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))))
63, 4, 5mp2an 683 . . . . . 6 (♯‘{𝑓𝑓:(1...𝐾)–1-1→(1...𝑁)}) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
72, 6eqtri 2787 . . . . 5 (♯‘𝑇) = ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾))))
8 elfznn0 12640 . . . . . . . . 9 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
98adantl 473 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ0)
10 hashfz1 13338 . . . . . . . 8 (𝐾 ∈ ℕ0 → (♯‘(1...𝐾)) = 𝐾)
119, 10syl 17 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = 𝐾)
1211fveq2d 6379 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(♯‘(1...𝐾))) = (!‘𝐾))
13 nnnn0 11546 . . . . . . . . 9 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
14 hashfz1 13338 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
1513, 14syl 17 . . . . . . . 8 (𝑁 ∈ ℕ → (♯‘(1...𝑁)) = 𝑁)
1615adantr 472 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝑁)) = 𝑁)
1716, 11oveq12d 6860 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))C(♯‘(1...𝐾))) = (𝑁C𝐾))
1812, 17oveq12d 6860 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘(♯‘(1...𝐾))) · ((♯‘(1...𝑁))C(♯‘(1...𝐾)))) = ((!‘𝐾) · (𝑁C𝐾)))
197, 18syl5eq 2811 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = ((!‘𝐾) · (𝑁C𝐾)))
2013adantr 472 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
21 faccl 13274 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
2220, 21syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
2322nncnd 11292 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
24 fznn0sub 12580 . . . . . . . . . 10 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ ℕ0)
2524adantl 473 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℕ0)
26 faccl 13274 . . . . . . . . 9 ((𝑁𝐾) ∈ ℕ0 → (!‘(𝑁𝐾)) ∈ ℕ)
2725, 26syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℕ)
2827nncnd 11292 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℂ)
2927nnne0d 11322 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ≠ 0)
3023, 28, 29divcld 11055 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁𝐾))) ∈ ℂ)
31 faccl 13274 . . . . . . . 8 (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ)
329, 31syl 17 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
3332nncnd 11292 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
3432nnne0d 11322 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ≠ 0)
3530, 33, 34divcan2d 11057 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾))) = ((!‘𝑁) / (!‘(𝑁𝐾))))
36 bcval2 13296 . . . . . . . 8 (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3736adantl 473 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3823, 28, 33, 29, 34divdiv1d 11086 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾)) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3937, 38eqtr4d 2802 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾)))
4039oveq2d 6858 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = ((!‘𝐾) · (((!‘𝑁) / (!‘(𝑁𝐾))) / (!‘𝐾))))
41 fzfid 12980 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
42 elfznn 12577 . . . . . . . . . 10 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
4342adantl 473 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
44 nnrp 12041 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
4544relogcld 24660 . . . . . . . . . 10 (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℝ)
4645recnd 10322 . . . . . . . . 9 (𝑛 ∈ ℕ → (log‘𝑛) ∈ ℂ)
4743, 46syl 17 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℂ)
4841, 47fsumcl 14749 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) ∈ ℂ)
49 fzfid 12980 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...(𝑁𝐾)) ∈ Fin)
50 elfznn 12577 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑁𝐾)) → 𝑛 ∈ ℕ)
5150adantl 473 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...(𝑁𝐾))) → 𝑛 ∈ ℕ)
5251, 46syl 17 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...(𝑁𝐾))) → (log‘𝑛) ∈ ℂ)
5349, 52fsumcl 14749 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) ∈ ℂ)
54 efsub 15112 . . . . . . 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))))
5548, 53, 54syl2anc 579 . . . . . 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))))
5625nn0red 11599 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℝ)
5756ltp1d 11208 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) < ((𝑁𝐾) + 1))
58 fzdisj 12575 . . . . . . . . . . 11 ((𝑁𝐾) < ((𝑁𝐾) + 1) → ((1...(𝑁𝐾)) ∩ (((𝑁𝐾) + 1)...𝑁)) = ∅)
5957, 58syl 17 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1...(𝑁𝐾)) ∩ (((𝑁𝐾) + 1)...𝑁)) = ∅)
60 fznn0sub2 12654 . . . . . . . . . . . . . . . 16 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ (0...𝑁))
6160adantl 473 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ (0...𝑁))
62 elfzle2 12552 . . . . . . . . . . . . . . 15 ((𝑁𝐾) ∈ (0...𝑁) → (𝑁𝐾) ≤ 𝑁)
6361, 62syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ≤ 𝑁)
6463adantr 472 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ≤ 𝑁)
65 simpr 477 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ ℕ)
66 nnuz 11923 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6765, 66syl6eleq 2854 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ (ℤ‘1))
68 nnz 11646 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
6968ad2antrr 717 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → 𝑁 ∈ ℤ)
70 elfz5 12541 . . . . . . . . . . . . . 14 (((𝑁𝐾) ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → ((𝑁𝐾) ∈ (1...𝑁) ↔ (𝑁𝐾) ≤ 𝑁))
7167, 69, 70syl2anc 579 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → ((𝑁𝐾) ∈ (1...𝑁) ↔ (𝑁𝐾) ≤ 𝑁))
7264, 71mpbird 248 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (𝑁𝐾) ∈ (1...𝑁))
73 fzsplit 12574 . . . . . . . . . . . 12 ((𝑁𝐾) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
7472, 73syl 17 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) ∈ ℕ) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
75 simpr 477 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (𝑁𝐾) = 0)
7675oveq2d 6858 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...(𝑁𝐾)) = (1...0))
77 fz10 12569 . . . . . . . . . . . . . 14 (1...0) = ∅
7876, 77syl6eq 2815 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...(𝑁𝐾)) = ∅)
7978uneq1d 3928 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)) = (∅ ∪ (((𝑁𝐾) + 1)...𝑁)))
80 uncom 3919 . . . . . . . . . . . . . 14 (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = ((((𝑁𝐾) + 1)...𝑁) ∪ ∅)
81 un0 4129 . . . . . . . . . . . . . 14 ((((𝑁𝐾) + 1)...𝑁) ∪ ∅) = (((𝑁𝐾) + 1)...𝑁)
8280, 81eqtri 2787 . . . . . . . . . . . . 13 (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = (((𝑁𝐾) + 1)...𝑁)
8375oveq1d 6857 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((𝑁𝐾) + 1) = (0 + 1))
84 1e0p1 11783 . . . . . . . . . . . . . . 15 1 = (0 + 1)
8583, 84syl6eqr 2817 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → ((𝑁𝐾) + 1) = 1)
8685oveq1d 6857 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (((𝑁𝐾) + 1)...𝑁) = (1...𝑁))
8782, 86syl5eq 2811 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (∅ ∪ (((𝑁𝐾) + 1)...𝑁)) = (1...𝑁))
8879, 87eqtr2d 2800 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑁𝐾) = 0) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
89 elnn0 11540 . . . . . . . . . . . 12 ((𝑁𝐾) ∈ ℕ0 ↔ ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
9025, 89sylib 209 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
9174, 88, 90mpjaodan 981 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝑁) = ((1...(𝑁𝐾)) ∪ (((𝑁𝐾) + 1)...𝑁)))
9259, 91, 41, 47fsumsplit 14756 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛) + Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
9392oveq1d 6857 . . . . . . . 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)))
94 fzfid 12980 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁𝐾) + 1)...𝑁) ∈ Fin)
95 nn0p1nn 11579 . . . . . . . . . . . . 13 ((𝑁𝐾) ∈ ℕ0 → ((𝑁𝐾) + 1) ∈ ℕ)
9625, 95syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℕ)
97 elfzuz 12545 . . . . . . . . . . . 12 (𝑛 ∈ (((𝑁𝐾) + 1)...𝑁) → 𝑛 ∈ (ℤ‘((𝑁𝐾) + 1)))
98 eluznn 11959 . . . . . . . . . . . 12 ((((𝑁𝐾) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((𝑁𝐾) + 1))) → 𝑛 ∈ ℕ)
9996, 97, 98syl2an 589 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑛 ∈ ℕ)
10099, 46syl 17 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘𝑛) ∈ ℂ)
10194, 100fsumcl 14749 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) ∈ ℂ)
10253, 101pncan2d 10648 . . . . . . . 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))
10393, 102eqtr2d 2800 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) = (Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
104103fveq2d 6379 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) = (exp‘(Σ𝑛 ∈ (1...𝑁)(log‘𝑛) − Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))))
10522nnne0d 11322 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ≠ 0)
106 eflog 24614 . . . . . . . . 9 (((!‘𝑁) ∈ ℂ ∧ (!‘𝑁) ≠ 0) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
10723, 105, 106syl2anc 579 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (!‘𝑁))
108 logfac 24638 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑛 ∈ (1...𝑁)(log‘𝑛))
10920, 108syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘(!‘𝑁)) = Σ𝑛 ∈ (1...𝑁)(log‘𝑛))
110109fveq2d 6379 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘𝑁))) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
111107, 110eqtr3d 2801 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)))
112 eflog 24614 . . . . . . . . 9 (((!‘(𝑁𝐾)) ∈ ℂ ∧ (!‘(𝑁𝐾)) ≠ 0) → (exp‘(log‘(!‘(𝑁𝐾)))) = (!‘(𝑁𝐾)))
11328, 29, 112syl2anc 579 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁𝐾)))) = (!‘(𝑁𝐾)))
114 logfac 24638 . . . . . . . . . 10 ((𝑁𝐾) ∈ ℕ0 → (log‘(!‘(𝑁𝐾))) = Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))
11525, 114syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘(!‘(𝑁𝐾))) = Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))
116115fveq2d 6379 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘(log‘(!‘(𝑁𝐾)))) = (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
117113, 116eqtr3d 2801 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) = (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛)))
118111, 117oveq12d 6860 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / (!‘(𝑁𝐾))) = ((exp‘Σ𝑛 ∈ (1...𝑁)(log‘𝑛)) / (exp‘Σ𝑛 ∈ (1...(𝑁𝐾))(log‘𝑛))))
11955, 104, 1183eqtr4d 2809 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) = ((!‘𝑁) / (!‘(𝑁𝐾))))
12035, 40, 1193eqtr4d 2809 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝐾) · (𝑁C𝐾)) = (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
12119, 120eqtrd 2799 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑇) = (exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)))
122 birthday.s . . . . . . . 8 𝑆 = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}
123 mapvalg 8070 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝐾)) = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)})
1244, 3, 123mp2an 683 . . . . . . . 8 ((1...𝑁) ↑𝑚 (1...𝐾)) = {𝑓𝑓:(1...𝐾)⟶(1...𝑁)}
125122, 124eqtr4i 2790 . . . . . . 7 𝑆 = ((1...𝑁) ↑𝑚 (1...𝐾))
126125fveq2i 6378 . . . . . 6 (♯‘𝑆) = (♯‘((1...𝑁) ↑𝑚 (1...𝐾)))
127 hashmap 13423 . . . . . . 7 (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘((1...𝑁) ↑𝑚 (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾))))
1284, 3, 127mp2an 683 . . . . . 6 (♯‘((1...𝑁) ↑𝑚 (1...𝐾))) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
129126, 128eqtri 2787 . . . . 5 (♯‘𝑆) = ((♯‘(1...𝑁))↑(♯‘(1...𝐾)))
13016, 11oveq12d 6860 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(1...𝑁))↑(♯‘(1...𝐾))) = (𝑁𝐾))
131129, 130syl5eq 2811 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (𝑁𝐾))
132 nncn 11283 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
133132adantr 472 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
134 nnne0 11310 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
135134adantr 472 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ≠ 0)
136 elfzelz 12549 . . . . . 6 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
137136adantl 473 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
138 explog 24631 . . . . 5 ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0 ∧ 𝐾 ∈ ℤ) → (𝑁𝐾) = (exp‘(𝐾 · (log‘𝑁))))
139133, 135, 137, 138syl3anc 1490 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) = (exp‘(𝐾 · (log‘𝑁))))
140131, 139eqtrd 2799 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘𝑆) = (exp‘(𝐾 · (log‘𝑁))))
141121, 140oveq12d 6860 . 2 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = ((exp‘Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛)) / (exp‘(𝐾 · (log‘𝑁)))))
1429nn0cnd 11600 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℂ)
143 nnrp 12041 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
144143adantr 472 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ+)
145144relogcld 24660 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℝ)
146145recnd 10322 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (log‘𝑁) ∈ ℂ)
147142, 146mulcld 10314 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 · (log‘𝑁)) ∈ ℂ)
148 efsub 15112 . . 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)))))
149101, 147, 148syl2anc 579 . 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)))))
150 relogdiv 24630 . . . . . . 7 ((𝑛 ∈ ℝ+𝑁 ∈ ℝ+) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
15144, 144, 150syl2anr 590 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ ℕ) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
15299, 151syldan 585 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) = ((log‘𝑛) − (log‘𝑁)))
153152sumeq2dv 14718 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)))
15468adantr 472 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
15525nn0zd 11727 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℤ)
156155peano2zd 11732 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℤ)
15799, 44syl 17 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑛 ∈ ℝ+)
158144adantr 472 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → 𝑁 ∈ ℝ+)
159157, 158rpdivcld 12087 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (𝑛 / 𝑁) ∈ ℝ+)
160159relogcld 24660 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℝ)
161160recnd 10322 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘(𝑛 / 𝑁)) ∈ ℂ)
162 fvoveq1 6865 . . . . . 6 (𝑛 = (𝑁𝑘) → (log‘(𝑛 / 𝑁)) = (log‘((𝑁𝑘) / 𝑁)))
163154, 156, 154, 161, 162fsumrev 14795 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1)))(log‘((𝑁𝑘) / 𝑁)))
164133subidd 10634 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝑁) = 0)
165 1cnd 10288 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℂ)
166133, 142, 165subsubd 10674 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝐾 − 1)) = ((𝑁𝐾) + 1))
167166oveq2d 6858 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝑁 − ((𝑁𝐾) + 1)))
168 ax-1cn 10247 . . . . . . . . . 10 1 ∈ ℂ
169 subcl 10534 . . . . . . . . . 10 ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐾 − 1) ∈ ℂ)
170142, 168, 169sylancl 580 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 − 1) ∈ ℂ)
171133, 170nncand 10651 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − (𝑁 − (𝐾 − 1))) = (𝐾 − 1))
172167, 171eqtr3d 2801 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − ((𝑁𝐾) + 1)) = (𝐾 − 1))
173164, 172oveq12d 6860 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1))) = (0...(𝐾 − 1)))
174133adantr 472 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
175 elfznn0 12640 . . . . . . . . . . 11 (𝑘 ∈ (0...(𝐾 − 1)) → 𝑘 ∈ ℕ0)
176175adantl 473 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℕ0)
177176nn0cnd 11600 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑘 ∈ ℂ)
178135adantr 472 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → 𝑁 ≠ 0)
179174, 177, 174, 178divsubdird 11094 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁𝑘) / 𝑁) = ((𝑁 / 𝑁) − (𝑘 / 𝑁)))
180174, 178dividd 11053 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (𝑁 / 𝑁) = 1)
181180oveq1d 6857 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁 / 𝑁) − (𝑘 / 𝑁)) = (1 − (𝑘 / 𝑁)))
182179, 181eqtrd 2799 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → ((𝑁𝑘) / 𝑁) = (1 − (𝑘 / 𝑁)))
183182fveq2d 6379 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝐾 − 1))) → (log‘((𝑁𝑘) / 𝑁)) = (log‘(1 − (𝑘 / 𝑁))))
184173, 183sumeq12rdv 14723 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑘 ∈ ((𝑁𝑁)...(𝑁 − ((𝑁𝐾) + 1)))(log‘((𝑁𝑘) / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
185163, 184eqtrd 2799 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘(𝑛 / 𝑁)) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
186146adantr 472 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)) → (log‘𝑁) ∈ ℂ)
18794, 100, 186fsumsub 14804 . . . . 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)))
188 fsumconst 14806 . . . . . . . 8 (((((𝑁𝐾) + 1)...𝑁) ∈ Fin ∧ (log‘𝑁) ∈ ℂ) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)))
18994, 146, 188syl2anc 579 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)))
190 1zzd 11655 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℤ)
191 fzen 12565 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (𝑁𝐾) ∈ ℤ) → (1...𝐾) ≈ ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))))
192190, 137, 155, 191syl3anc 1490 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))))
19325nn0cnd 11600 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℂ)
194 addcom 10476 . . . . . . . . . . . . 13 ((1 ∈ ℂ ∧ (𝑁𝐾) ∈ ℂ) → (1 + (𝑁𝐾)) = ((𝑁𝐾) + 1))
195168, 193, 194sylancr 581 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1 + (𝑁𝐾)) = ((𝑁𝐾) + 1))
196142, 133pncan3d 10649 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 + (𝑁𝐾)) = 𝑁)
197195, 196oveq12d 6860 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((1 + (𝑁𝐾))...(𝐾 + (𝑁𝐾))) = (((𝑁𝐾) + 1)...𝑁))
198192, 197breqtrd 4835 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (1...𝐾) ≈ (((𝑁𝐾) + 1)...𝑁))
199 hasheni 13340 . . . . . . . . . 10 ((1...𝐾) ≈ (((𝑁𝐾) + 1)...𝑁) → (♯‘(1...𝐾)) = (♯‘(((𝑁𝐾) + 1)...𝑁)))
200198, 199syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(1...𝐾)) = (♯‘(((𝑁𝐾) + 1)...𝑁)))
201200, 11eqtr3d 2801 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (♯‘(((𝑁𝐾) + 1)...𝑁)) = 𝐾)
202201oveq1d 6857 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘(((𝑁𝐾) + 1)...𝑁)) · (log‘𝑁)) = (𝐾 · (log‘𝑁)))
203189, 202eqtrd 2799 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑁) = (𝐾 · (log‘𝑁)))
204203oveq2d 6858 . . . . 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))))
205187, 204eqtrd 2799 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)((log‘𝑛) − (log‘𝑁)) = (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))))
206153, 185, 2053eqtr3rd 2808 . . 3 ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → (Σ𝑛 ∈ (((𝑁𝐾) + 1)...𝑁)(log‘𝑛) − (𝐾 · (log‘𝑁))) = Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁))))
207206fveq2d 6379 . 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)))))
208141, 149, 2073eqtr2d 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 197  wa 384  wo 873   = wceq 1652  wcel 2155  {cab 2751  wne 2937  cun 3730  cin 3731  c0 4079   class class class wbr 4809  wf 6064  1-1wf1 6065  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  cen 8157  Fincfn 8160  cc 10187  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194   < clt 10328  cle 10329  cmin 10520   / cdiv 10938  cn 11274  0cn0 11538  cz 11624  cuz 11886  +crp 12028  ...cfz 12533  cexp 13067  !cfa 13264  Ccbc 13293  chash 13321  Σcsu 14701  expce 15074  logclog 24592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-dec 11741  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ioc 12382  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-shft 14092  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-limsup 14487  df-clim 14504  df-rlim 14505  df-sum 14702  df-ef 15080  df-sin 15082  df-cos 15083  df-pi 15085  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-starv 16229  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-unif 16237  df-hom 16238  df-cco 16239  df-rest 16349  df-topn 16350  df-0g 16368  df-gsum 16369  df-topgen 16370  df-pt 16371  df-prds 16374  df-xrs 16428  df-qtop 16433  df-imas 16434  df-xps 16436  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-mulg 17808  df-cntz 18013  df-cmn 18461  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-fbas 20016  df-fg 20017  df-cnfld 20020  df-top 20978  df-topon 20995  df-topsp 21017  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-lp 21220  df-perf 21221  df-cn 21311  df-cnp 21312  df-haus 21399  df-tx 21645  df-hmeo 21838  df-fil 21929  df-fm 22021  df-flim 22022  df-flf 22023  df-xms 22404  df-ms 22405  df-tms 22406  df-cncf 22960  df-limc 23921  df-dv 23922  df-log 24594
This theorem is referenced by:  birthdaylem3  24971
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