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Theorem bcval5 10477
Description: Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
Assertion
Ref Expression
bcval5 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))

Proof of Theorem bcval5
Dummy variables 𝑥 𝑘 𝑦 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcval2 10464 . . . 4 (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
21adantl 275 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3 simprl 505 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → 𝑘 ∈ ℂ)
4 simprr 506 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → 𝑥 ∈ ℂ)
53, 4mulcld 7754 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
6 simpr1 972 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑘 ∈ ℂ)
7 simpr2 973 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
8 simpr3 974 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
96, 7, 8mulassd 7757 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 · 𝑥) · 𝑦) = (𝑘 · (𝑥 · 𝑦)))
10 simpll 503 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
1110nn0zd 9139 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
12 simplr 504 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ)
1312nnzd 9140 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
1411, 13zsubcld 9146 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℤ)
1514peano2zd 9144 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℤ)
16 1red 7749 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℝ)
1712nnred 8701 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℝ)
1810nn0red 8999 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
1912nnge1d 8731 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 1 ≤ 𝐾)
2016, 17, 18, 19lesub2dd 8292 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ≤ (𝑁 − 1))
2114zred 9141 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℝ)
22 leaddsub 8168 . . . . . . . . . . . 12 (((𝑁𝐾) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝑁𝐾) + 1) ≤ 𝑁 ↔ (𝑁𝐾) ≤ (𝑁 − 1)))
2321, 16, 18, 22syl3anc 1201 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁𝐾) + 1) ≤ 𝑁 ↔ (𝑁𝐾) ≤ (𝑁 − 1)))
2420, 23mpbird 166 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ≤ 𝑁)
25 eluz2 9300 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘((𝑁𝐾) + 1)) ↔ (((𝑁𝐾) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑁𝐾) + 1) ≤ 𝑁))
2615, 11, 24, 25syl3anbrc 1150 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ‘((𝑁𝐾) + 1)))
2726adantrr 470 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → 𝑁 ∈ (ℤ‘((𝑁𝐾) + 1)))
28 simprr 506 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (𝑁𝐾) ∈ ℕ)
29 nnuz 9329 . . . . . . . . 9 ℕ = (ℤ‘1)
3028, 29eleqtrdi 2210 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (𝑁𝐾) ∈ (ℤ‘1))
31 fvi 5446 . . . . . . . . . 10 (𝑘 ∈ V → ( I ‘𝑘) = 𝑘)
3231elv 2664 . . . . . . . . 9 ( I ‘𝑘) = 𝑘
33 eluzelcn 9305 . . . . . . . . . 10 (𝑘 ∈ (ℤ‘1) → 𝑘 ∈ ℂ)
3433adantl 275 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ 𝑘 ∈ (ℤ‘1)) → 𝑘 ∈ ℂ)
3532, 34eqeltrid 2204 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ 𝑘 ∈ (ℤ‘1)) → ( I ‘𝑘) ∈ ℂ)
365, 9, 27, 30, 35seq3split 10220 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (seq1( · , I )‘𝑁) = ((seq1( · , I )‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)))
37 elfzuz3 9771 . . . . . . . . . . 11 (𝐾 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝐾))
3837adantl 275 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ𝐾))
39 eluznn 9362 . . . . . . . . . 10 ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝐾)) → 𝑁 ∈ ℕ)
4012, 38, 39syl2anc 408 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ)
4140adantrr 470 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → 𝑁 ∈ ℕ)
42 facnn 10441 . . . . . . . 8 (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁))
4341, 42syl 14 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (!‘𝑁) = (seq1( · , I )‘𝑁))
44 facnn 10441 . . . . . . . . 9 ((𝑁𝐾) ∈ ℕ → (!‘(𝑁𝐾)) = (seq1( · , I )‘(𝑁𝐾)))
4528, 44syl 14 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (!‘(𝑁𝐾)) = (seq1( · , I )‘(𝑁𝐾)))
4645oveq1d 5757 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)) = ((seq1( · , I )‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)))
4736, 43, 463eqtr4d 2160 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)))
4847expr 372 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) ∈ ℕ → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁))))
4910faccld 10450 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
5049nncnd 8702 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
5150mulid2d 7752 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (1 · (!‘𝑁)) = (!‘𝑁))
5240, 42syl 14 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (seq1( · , I )‘𝑁))
5352oveq2d 5758 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (1 · (!‘𝑁)) = (1 · (seq1( · , I )‘𝑁)))
5451, 53eqtr3d 2152 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (1 · (seq1( · , I )‘𝑁)))
55 fveq2 5389 . . . . . . . . 9 ((𝑁𝐾) = 0 → (!‘(𝑁𝐾)) = (!‘0))
56 fac0 10442 . . . . . . . . 9 (!‘0) = 1
5755, 56syl6eq 2166 . . . . . . . 8 ((𝑁𝐾) = 0 → (!‘(𝑁𝐾)) = 1)
58 oveq1 5749 . . . . . . . . . . 11 ((𝑁𝐾) = 0 → ((𝑁𝐾) + 1) = (0 + 1))
59 0p1e1 8802 . . . . . . . . . . 11 (0 + 1) = 1
6058, 59syl6eq 2166 . . . . . . . . . 10 ((𝑁𝐾) = 0 → ((𝑁𝐾) + 1) = 1)
6160seqeq1d 10192 . . . . . . . . 9 ((𝑁𝐾) = 0 → seq((𝑁𝐾) + 1)( · , I ) = seq1( · , I ))
6261fveq1d 5391 . . . . . . . 8 ((𝑁𝐾) = 0 → (seq((𝑁𝐾) + 1)( · , I )‘𝑁) = (seq1( · , I )‘𝑁))
6357, 62oveq12d 5760 . . . . . . 7 ((𝑁𝐾) = 0 → ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)) = (1 · (seq1( · , I )‘𝑁)))
6463eqeq2d 2129 . . . . . 6 ((𝑁𝐾) = 0 → ((!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)) ↔ (!‘𝑁) = (1 · (seq1( · , I )‘𝑁))))
6554, 64syl5ibrcom 156 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) = 0 → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁))))
66 fznn0sub 9805 . . . . . . 7 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ ℕ0)
6766adantl 275 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℕ0)
68 elnn0 8947 . . . . . 6 ((𝑁𝐾) ∈ ℕ0 ↔ ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
6967, 68sylib 121 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
7048, 65, 69mpjaod 692 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)))
7170oveq1d 5757 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))) = (((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
72 eqid 2117 . . . . . 6 (ℤ‘((𝑁𝐾) + 1)) = (ℤ‘((𝑁𝐾) + 1))
73 fvi 5446 . . . . . . . 8 (𝑓 ∈ V → ( I ‘𝑓) = 𝑓)
7473elv 2664 . . . . . . 7 ( I ‘𝑓) = 𝑓
75 eluzelcn 9305 . . . . . . . 8 (𝑓 ∈ (ℤ‘((𝑁𝐾) + 1)) → 𝑓 ∈ ℂ)
7675adantl 275 . . . . . . 7 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑓 ∈ (ℤ‘((𝑁𝐾) + 1))) → 𝑓 ∈ ℂ)
7774, 76eqeltrid 2204 . . . . . 6 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑓 ∈ (ℤ‘((𝑁𝐾) + 1))) → ( I ‘𝑓) ∈ ℂ)
78 mulcl 7715 . . . . . . 7 ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ) → (𝑓 · 𝑔) ∈ ℂ)
7978adantl 275 . . . . . 6 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ)) → (𝑓 · 𝑔) ∈ ℂ)
8072, 15, 77, 79seqf 10202 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → seq((𝑁𝐾) + 1)( · , I ):(ℤ‘((𝑁𝐾) + 1))⟶ℂ)
8180, 26ffvelrnd 5524 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (seq((𝑁𝐾) + 1)( · , I )‘𝑁) ∈ ℂ)
8212nnnn0d 8998 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ0)
8382faccld 10450 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
8483nncnd 8702 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
8567faccld 10450 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℕ)
8685nncnd 8702 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℂ)
8783nnap0d 8734 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) # 0)
8885nnap0d 8734 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) # 0)
8981, 84, 86, 87, 88divcanap5d 8545 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I )‘𝑁)) / ((!‘(𝑁𝐾)) · (!‘𝐾))) = ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
902, 71, 893eqtrd 2154 . 2 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
91 simplr 504 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ)
9291nnnn0d 8998 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ0)
9392faccld 10450 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
9493nncnd 8702 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
9593nnap0d 8734 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) # 0)
9694, 95div0apd 8515 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (0 / (!‘𝐾)) = 0)
97 mulcl 7715 . . . . . 6 ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
9897adantl 275 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
99 eluzelcn 9305 . . . . . . 7 (𝑘 ∈ (ℤ‘((𝑁𝐾) + 1)) → 𝑘 ∈ ℂ)
10099adantl 275 . . . . . 6 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (ℤ‘((𝑁𝐾) + 1))) → 𝑘 ∈ ℂ)
10132, 100eqeltrid 2204 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (ℤ‘((𝑁𝐾) + 1))) → ( I ‘𝑘) ∈ ℂ)
102 simpr 109 . . . . . 6 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ ℂ) → 𝑘 ∈ ℂ)
103102mul02d 8122 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ ℂ) → (0 · 𝑘) = 0)
104102mul01d 8123 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ ℂ) → (𝑘 · 0) = 0)
105 simpr 109 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ¬ 𝐾 ∈ (0...𝑁))
106 nn0uz 9328 . . . . . . . . . . . 12 0 = (ℤ‘0)
10792, 106eleqtrdi 2210 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ (ℤ‘0))
108 simpll 503 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
109108nn0zd 9139 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
110 elfz5 9766 . . . . . . . . . . 11 ((𝐾 ∈ (ℤ‘0) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...𝑁) ↔ 𝐾𝑁))
111107, 109, 110syl2anc 408 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝐾 ∈ (0...𝑁) ↔ 𝐾𝑁))
112 nn0re 8954 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
113112ad2antrr 479 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
114 nnre 8695 . . . . . . . . . . . 12 (𝐾 ∈ ℕ → 𝐾 ∈ ℝ)
115114ad2antlr 480 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℝ)
116113, 115subge0d 8265 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (0 ≤ (𝑁𝐾) ↔ 𝐾𝑁))
117111, 116bitr4d 190 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝐾 ∈ (0...𝑁) ↔ 0 ≤ (𝑁𝐾)))
118105, 117mtbid 646 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ¬ 0 ≤ (𝑁𝐾))
119 simpl 108 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝑁 ∈ ℕ0)
120119nn0zd 9139 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝑁 ∈ ℤ)
121 simpr 109 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝐾 ∈ ℕ)
122121nnzd 9140 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝐾 ∈ ℤ)
123120, 122zsubcld 9146 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁𝐾) ∈ ℤ)
124123adantr 274 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℤ)
125 0z 9033 . . . . . . . . 9 0 ∈ ℤ
126 zltnle 9068 . . . . . . . . 9 (((𝑁𝐾) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑁𝐾) < 0 ↔ ¬ 0 ≤ (𝑁𝐾)))
127124, 125, 126sylancl 409 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) < 0 ↔ ¬ 0 ≤ (𝑁𝐾)))
128118, 127mpbird 166 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) < 0)
129 zltp1le 9076 . . . . . . . 8 (((𝑁𝐾) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑁𝐾) < 0 ↔ ((𝑁𝐾) + 1) ≤ 0))
130124, 125, 129sylancl 409 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) < 0 ↔ ((𝑁𝐾) + 1) ≤ 0))
131128, 130mpbid 146 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ≤ 0)
132 nn0ge0 8970 . . . . . . 7 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
133132ad2antrr 479 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 0 ≤ 𝑁)
134 0zd 9034 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 0 ∈ ℤ)
135124peano2zd 9144 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℤ)
136 elfz 9764 . . . . . . 7 ((0 ∈ ℤ ∧ ((𝑁𝐾) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∈ (((𝑁𝐾) + 1)...𝑁) ↔ (((𝑁𝐾) + 1) ≤ 0 ∧ 0 ≤ 𝑁)))
137134, 135, 109, 136syl3anc 1201 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (0 ∈ (((𝑁𝐾) + 1)...𝑁) ↔ (((𝑁𝐾) + 1) ≤ 0 ∧ 0 ≤ 𝑁)))
138131, 133, 137mpbir2and 913 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 0 ∈ (((𝑁𝐾) + 1)...𝑁))
139 0cn 7726 . . . . . 6 0 ∈ ℂ
140 fvi 5446 . . . . . 6 (0 ∈ ℂ → ( I ‘0) = 0)
141139, 140mp1i 10 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ( I ‘0) = 0)
14298, 101, 103, 104, 138, 141seq3z 10252 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (seq((𝑁𝐾) + 1)( · , I )‘𝑁) = 0)
143142oveq1d 5757 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)) = (0 / (!‘𝐾)))
144 nnz 9041 . . . . 5 (𝐾 ∈ ℕ → 𝐾 ∈ ℤ)
145 bcval3 10465 . . . . 5 ((𝑁 ∈ ℕ0𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
146144, 145syl3an2 1235 . . . 4 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
1471463expa 1166 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
14896, 143, 1473eqtr4rd 2161 . 2 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
149 0zd 9034 . . . 4 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 0 ∈ ℤ)
150 fzdcel 9788 . . . 4 ((𝐾 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (0...𝑁))
151122, 149, 120, 150syl3anc 1201 . . 3 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → DECID 𝐾 ∈ (0...𝑁))
152 exmiddc 806 . . 3 (DECID 𝐾 ∈ (0...𝑁) → (𝐾 ∈ (0...𝑁) ∨ ¬ 𝐾 ∈ (0...𝑁)))
153151, 152syl 14 . 2 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝐾 ∈ (0...𝑁) ∨ ¬ 𝐾 ∈ (0...𝑁)))
15490, 148, 153mpjaodan 772 1 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682  DECID wdc 804  w3a 947   = wceq 1316  wcel 1465  Vcvv 2660   class class class wbr 3899   I cid 4180  cfv 5093  (class class class)co 5742  cc 7586  cr 7587  0cc0 7588  1c1 7589   + caddc 7591   · cmul 7593   < clt 7768  cle 7769  cmin 7901   / cdiv 8400  cn 8688  0cn0 8945  cz 9022  cuz 9294  ...cfz 9758  seqcseq 10186  !cfa 10439  Ccbc 10461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-fz 9759  df-seqfrec 10187  df-fac 10440  df-bc 10462
This theorem is referenced by:  bcn2  10478
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