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Theorem ibcval5 10006
 Description: Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁 − 𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Jim Kingdon, 6-Nov-2021.)
Assertion
Ref Expression
ibcval5 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) / (!‘𝐾)))

Proof of Theorem ibcval5
Dummy variables 𝑥 𝑘 𝑦 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcval2 9993 . . . 4 (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
21adantl 271 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
3 simprl 498 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → 𝑘 ∈ ℂ)
4 simprr 499 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → 𝑥 ∈ ℂ)
53, 4mulcld 7411 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
6 simpr1 945 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑘 ∈ ℂ)
7 simpr2 946 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
8 simpr3 947 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
96, 7, 8mulassd 7414 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 · 𝑥) · 𝑦) = (𝑘 · (𝑥 · 𝑦)))
10 simpll 496 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
1110nn0zd 8762 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
12 simplr 497 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ)
1312nnzd 8763 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
1411, 13zsubcld 8769 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℤ)
1514peano2zd 8767 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℤ)
16 1red 7406 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 1 ∈ ℝ)
1712nnred 8329 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℝ)
1810nn0red 8619 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
1912nnge1d 8358 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 1 ≤ 𝐾)
2016, 17, 18, 19lesub2dd 7939 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ≤ (𝑁 − 1))
2114zred 8764 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℝ)
22 leaddsub 7819 . . . . . . . . . . . 12 (((𝑁𝐾) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝑁𝐾) + 1) ≤ 𝑁 ↔ (𝑁𝐾) ≤ (𝑁 − 1)))
2321, 16, 18, 22syl3anc 1170 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (((𝑁𝐾) + 1) ≤ 𝑁 ↔ (𝑁𝐾) ≤ (𝑁 − 1)))
2420, 23mpbird 165 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ≤ 𝑁)
25 eluz2 8920 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘((𝑁𝐾) + 1)) ↔ (((𝑁𝐾) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑁𝐾) + 1) ≤ 𝑁))
2615, 11, 24, 25syl3anbrc 1123 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ‘((𝑁𝐾) + 1)))
2726adantrr 463 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → 𝑁 ∈ (ℤ‘((𝑁𝐾) + 1)))
28 cnex 7369 . . . . . . . . 9 ℂ ∈ V
2928a1i 9 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → ℂ ∈ V)
30 simprr 499 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (𝑁𝐾) ∈ ℕ)
31 nnuz 8949 . . . . . . . . 9 ℕ = (ℤ‘1)
3230, 31syl6eleq 2175 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (𝑁𝐾) ∈ (ℤ‘1))
33 vex 2615 . . . . . . . . . 10 𝑘 ∈ V
34 fvi 5306 . . . . . . . . . 10 (𝑘 ∈ V → ( I ‘𝑘) = 𝑘)
3533, 34ax-mp 7 . . . . . . . . 9 ( I ‘𝑘) = 𝑘
36 eluzelcn 8925 . . . . . . . . . 10 (𝑘 ∈ (ℤ‘1) → 𝑘 ∈ ℂ)
3736adantl 271 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ 𝑘 ∈ (ℤ‘1)) → 𝑘 ∈ ℂ)
3835, 37syl5eqel 2169 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) ∧ 𝑘 ∈ (ℤ‘1)) → ( I ‘𝑘) ∈ ℂ)
395, 9, 27, 29, 32, 38iseqsplit 9773 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (seq1( · , I , ℂ)‘𝑁) = ((seq1( · , I , ℂ)‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)))
40 elfzuz3 9332 . . . . . . . . . . 11 (𝐾 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝐾))
4140adantl 271 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ𝐾))
42 eluznn 8982 . . . . . . . . . 10 ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝐾)) → 𝑁 ∈ ℕ)
4312, 41, 42syl2anc 403 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ)
4443adantrr 463 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → 𝑁 ∈ ℕ)
45 facnn 9970 . . . . . . . 8 (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I , ℂ)‘𝑁))
4644, 45syl 14 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (!‘𝑁) = (seq1( · , I , ℂ)‘𝑁))
47 facnn 9970 . . . . . . . . 9 ((𝑁𝐾) ∈ ℕ → (!‘(𝑁𝐾)) = (seq1( · , I , ℂ)‘(𝑁𝐾)))
4830, 47syl 14 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (!‘(𝑁𝐾)) = (seq1( · , I , ℂ)‘(𝑁𝐾)))
4948oveq1d 5606 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)) = ((seq1( · , I , ℂ)‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)))
5039, 46, 493eqtr4d 2125 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ (𝑁𝐾) ∈ ℕ)) → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)))
5150expr 367 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) ∈ ℕ → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁))))
5210faccld 9979 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℕ)
5352nncnd 8330 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) ∈ ℂ)
5453mulid2d 7409 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (1 · (!‘𝑁)) = (!‘𝑁))
5543, 45syl 14 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (seq1( · , I , ℂ)‘𝑁))
5655oveq2d 5607 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (1 · (!‘𝑁)) = (1 · (seq1( · , I , ℂ)‘𝑁)))
5754, 56eqtr3d 2117 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = (1 · (seq1( · , I , ℂ)‘𝑁)))
58 fveq2 5253 . . . . . . . . 9 ((𝑁𝐾) = 0 → (!‘(𝑁𝐾)) = (!‘0))
59 fac0 9971 . . . . . . . . 9 (!‘0) = 1
6058, 59syl6eq 2131 . . . . . . . 8 ((𝑁𝐾) = 0 → (!‘(𝑁𝐾)) = 1)
61 oveq1 5598 . . . . . . . . . . 11 ((𝑁𝐾) = 0 → ((𝑁𝐾) + 1) = (0 + 1))
62 0p1e1 8430 . . . . . . . . . . 11 (0 + 1) = 1
6361, 62syl6eq 2131 . . . . . . . . . 10 ((𝑁𝐾) = 0 → ((𝑁𝐾) + 1) = 1)
64 iseqeq1 9743 . . . . . . . . . 10 (((𝑁𝐾) + 1) = 1 → seq((𝑁𝐾) + 1)( · , I , ℂ) = seq1( · , I , ℂ))
6563, 64syl 14 . . . . . . . . 9 ((𝑁𝐾) = 0 → seq((𝑁𝐾) + 1)( · , I , ℂ) = seq1( · , I , ℂ))
6665fveq1d 5255 . . . . . . . 8 ((𝑁𝐾) = 0 → (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) = (seq1( · , I , ℂ)‘𝑁))
6760, 66oveq12d 5609 . . . . . . 7 ((𝑁𝐾) = 0 → ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)) = (1 · (seq1( · , I , ℂ)‘𝑁)))
6867eqeq2d 2094 . . . . . 6 ((𝑁𝐾) = 0 → ((!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)) ↔ (!‘𝑁) = (1 · (seq1( · , I , ℂ)‘𝑁))))
6957, 68syl5ibrcom 155 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) = 0 → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁))))
70 fznn0sub 9365 . . . . . . 7 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ ℕ0)
7170adantl 271 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℕ0)
72 elnn0 8567 . . . . . 6 ((𝑁𝐾) ∈ ℕ0 ↔ ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
7371, 72sylib 120 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) ∈ ℕ ∨ (𝑁𝐾) = 0))
7451, 69, 73mpjaod 671 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝑁) = ((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)))
7574oveq1d 5606 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))) = (((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
76 vex 2615 . . . . . . 7 𝑓 ∈ V
77 fvi 5306 . . . . . . 7 (𝑓 ∈ V → ( I ‘𝑓) = 𝑓)
7876, 77ax-mp 7 . . . . . 6 ( I ‘𝑓) = 𝑓
79 eluzelcn 8925 . . . . . . 7 (𝑓 ∈ (ℤ‘((𝑁𝐾) + 1)) → 𝑓 ∈ ℂ)
8079adantl 271 . . . . . 6 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑓 ∈ (ℤ‘((𝑁𝐾) + 1))) → 𝑓 ∈ ℂ)
8178, 80syl5eqel 2169 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑓 ∈ (ℤ‘((𝑁𝐾) + 1))) → ( I ‘𝑓) ∈ ℂ)
82 mulcl 7372 . . . . . 6 ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ) → (𝑓 · 𝑔) ∈ ℂ)
8382adantl 271 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) ∧ (𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ)) → (𝑓 · 𝑔) ∈ ℂ)
8426, 81, 83iseqcl 9756 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) ∈ ℂ)
8512nnnn0d 8618 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ0)
8685faccld 9979 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
8786nncnd 8330 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
8871faccld 9979 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℕ)
8988nncnd 8330 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) ∈ ℂ)
9086nnap0d 8361 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) # 0)
9188nnap0d 8361 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (!‘(𝑁𝐾)) # 0)
9284, 87, 89, 90, 91divcanap5d 8180 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (((!‘(𝑁𝐾)) · (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁)) / ((!‘(𝑁𝐾)) · (!‘𝐾))) = ((seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) / (!‘𝐾)))
932, 75, 923eqtrd 2119 . 2 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) / (!‘𝐾)))
94 simplr 497 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ)
9594nnnn0d 8618 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℕ0)
9695faccld 9979 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℕ)
9796nncnd 8330 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) ∈ ℂ)
9896nnap0d 8361 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (!‘𝐾) # 0)
9997, 98div0apd 8152 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (0 / (!‘𝐾)) = 0)
100 mulcl 7372 . . . . . 6 ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
101100adantl 271 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
102 eluzelcn 8925 . . . . . . 7 (𝑘 ∈ (ℤ‘((𝑁𝐾) + 1)) → 𝑘 ∈ ℂ)
103102adantl 271 . . . . . 6 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (ℤ‘((𝑁𝐾) + 1))) → 𝑘 ∈ ℂ)
10435, 103syl5eqel 2169 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ (ℤ‘((𝑁𝐾) + 1))) → ( I ‘𝑘) ∈ ℂ)
10528a1i 9 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ℂ ∈ V)
106 simpr 108 . . . . . 6 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ ℂ) → 𝑘 ∈ ℂ)
107106mul02d 7773 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ ℂ) → (0 · 𝑘) = 0)
108106mul01d 7774 . . . . 5 ((((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) ∧ 𝑘 ∈ ℂ) → (𝑘 · 0) = 0)
109 simpr 108 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ¬ 𝐾 ∈ (0...𝑁))
110 nn0uz 8948 . . . . . . . . . . . 12 0 = (ℤ‘0)
11195, 110syl6eleq 2175 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ (ℤ‘0))
112 simpll 496 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
113112nn0zd 8762 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
114 elfz5 9327 . . . . . . . . . . 11 ((𝐾 ∈ (ℤ‘0) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...𝑁) ↔ 𝐾𝑁))
115111, 113, 114syl2anc 403 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝐾 ∈ (0...𝑁) ↔ 𝐾𝑁))
116 nn0re 8574 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
117116ad2antrr 472 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
118 nnre 8323 . . . . . . . . . . . 12 (𝐾 ∈ ℕ → 𝐾 ∈ ℝ)
119118ad2antlr 473 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ ℝ)
120117, 119subge0d 7912 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (0 ≤ (𝑁𝐾) ↔ 𝐾𝑁))
121115, 120bitr4d 189 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝐾 ∈ (0...𝑁) ↔ 0 ≤ (𝑁𝐾)))
122109, 121mtbid 630 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ¬ 0 ≤ (𝑁𝐾))
123 simpl 107 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝑁 ∈ ℕ0)
124123nn0zd 8762 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝑁 ∈ ℤ)
125 simpr 108 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝐾 ∈ ℕ)
126125nnzd 8763 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 𝐾 ∈ ℤ)
127124, 126zsubcld 8769 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁𝐾) ∈ ℤ)
128127adantr 270 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ ℤ)
129 0z 8657 . . . . . . . . 9 0 ∈ ℤ
130 zltnle 8692 . . . . . . . . 9 (((𝑁𝐾) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑁𝐾) < 0 ↔ ¬ 0 ≤ (𝑁𝐾)))
131128, 129, 130sylancl 404 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) < 0 ↔ ¬ 0 ≤ (𝑁𝐾)))
132122, 131mpbird 165 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) < 0)
133 zltp1le 8700 . . . . . . . 8 (((𝑁𝐾) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑁𝐾) < 0 ↔ ((𝑁𝐾) + 1) ≤ 0))
134128, 129, 133sylancl 404 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) < 0 ↔ ((𝑁𝐾) + 1) ≤ 0))
135132, 134mpbid 145 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ≤ 0)
136 nn0ge0 8590 . . . . . . 7 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
137136ad2antrr 472 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 0 ≤ 𝑁)
138 0zd 8658 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 0 ∈ ℤ)
139128peano2zd 8767 . . . . . . 7 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((𝑁𝐾) + 1) ∈ ℤ)
140 elfz 9325 . . . . . . 7 ((0 ∈ ℤ ∧ ((𝑁𝐾) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∈ (((𝑁𝐾) + 1)...𝑁) ↔ (((𝑁𝐾) + 1) ≤ 0 ∧ 0 ≤ 𝑁)))
141138, 139, 113, 140syl3anc 1170 . . . . . 6 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (0 ∈ (((𝑁𝐾) + 1)...𝑁) ↔ (((𝑁𝐾) + 1) ≤ 0 ∧ 0 ≤ 𝑁)))
142135, 137, 141mpbir2and 886 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 0 ∈ (((𝑁𝐾) + 1)...𝑁))
143 elex 2621 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ V)
144143ad2antrr 472 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 ∈ V)
145 0cn 7383 . . . . . 6 0 ∈ ℂ
146 fvi 5306 . . . . . 6 (0 ∈ ℂ → ( I ‘0) = 0)
147145, 146mp1i 10 . . . . 5 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ( I ‘0) = 0)
148101, 104, 105, 107, 108, 142, 144, 147iseqz 9785 . . . 4 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) = 0)
149148oveq1d 5606 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → ((seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) / (!‘𝐾)) = (0 / (!‘𝐾)))
150 nnz 8665 . . . . 5 (𝐾 ∈ ℕ → 𝐾 ∈ ℤ)
151 bcval3 9994 . . . . 5 ((𝑁 ∈ ℕ0𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
152150, 151syl3an2 1204 . . . 4 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
1531523expa 1139 . . 3 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
15499, 149, 1533eqtr4rd 2126 . 2 (((𝑁 ∈ ℕ0𝐾 ∈ ℕ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) / (!‘𝐾)))
155 0zd 8658 . . . 4 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → 0 ∈ ℤ)
156 fzdcel 9349 . . . 4 ((𝐾 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (0...𝑁))
157126, 155, 124, 156syl3anc 1170 . . 3 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → DECID 𝐾 ∈ (0...𝑁))
158 exmiddc 778 . . 3 (DECID 𝐾 ∈ (0...𝑁) → (𝐾 ∈ (0...𝑁) ∨ ¬ 𝐾 ∈ (0...𝑁)))
159157, 158syl 14 . 2 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝐾 ∈ (0...𝑁) ∨ ¬ 𝐾 ∈ (0...𝑁)))
16093, 154, 159mpjaodan 745 1 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I , ℂ)‘𝑁) / (!‘𝐾)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662  DECID wdc 776   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  Vcvv 2612   class class class wbr 3811   I cid 4079  ‘cfv 4969  (class class class)co 5591  ℂcc 7251  ℝcr 7252  0cc0 7253  1c1 7254   + caddc 7256   · cmul 7258   < clt 7425   ≤ cle 7426   − cmin 7556   / cdiv 8037  ℕcn 8316  ℕ0cn0 8565  ℤcz 8646  ℤ≥cuz 8914  ...cfz 9319  seqcseq 9740  !cfa 9968  Ccbc 9990 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-mulrcl 7347  ax-addcom 7348  ax-mulcom 7349  ax-addass 7350  ax-mulass 7351  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-1rid 7355  ax-0id 7356  ax-rnegex 7357  ax-precex 7358  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-apti 7363  ax-pre-ltadd 7364  ax-pre-mulgt0 7365  ax-pre-mulext 7366 This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-reap 7952  df-ap 7959  df-div 8038  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915  df-q 9000  df-fz 9320  df-iseq 9741  df-fac 9969  df-bc 9991 This theorem is referenced by:  bcn2  10007
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