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Theorem bcval 12953
 Description: Value of the binomial coefficient, 𝑁 choose 𝐾. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝐾 ≤ 𝑁 does not hold. See bcval2 12954 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
Assertion
Ref Expression
bcval ((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0))

Proof of Theorem bcval
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . 4 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
21eleq2d 2673 . . 3 (𝑛 = 𝑁 → (𝑘 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑁)))
3 fveq2 6103 . . . 4 (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
4 oveq1 6556 . . . . . 6 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
54fveq2d 6107 . . . . 5 (𝑛 = 𝑁 → (!‘(𝑛𝑘)) = (!‘(𝑁𝑘)))
65oveq1d 6564 . . . 4 (𝑛 = 𝑁 → ((!‘(𝑛𝑘)) · (!‘𝑘)) = ((!‘(𝑁𝑘)) · (!‘𝑘)))
73, 6oveq12d 6567 . . 3 (𝑛 = 𝑁 → ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))) = ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))))
82, 7ifbieq1d 4059 . 2 (𝑛 = 𝑁 → if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0) = if(𝑘 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))), 0))
9 eleq1 2676 . . 3 (𝑘 = 𝐾 → (𝑘 ∈ (0...𝑁) ↔ 𝐾 ∈ (0...𝑁)))
10 oveq2 6557 . . . . . 6 (𝑘 = 𝐾 → (𝑁𝑘) = (𝑁𝐾))
1110fveq2d 6107 . . . . 5 (𝑘 = 𝐾 → (!‘(𝑁𝑘)) = (!‘(𝑁𝐾)))
12 fveq2 6103 . . . . 5 (𝑘 = 𝐾 → (!‘𝑘) = (!‘𝐾))
1311, 12oveq12d 6567 . . . 4 (𝑘 = 𝐾 → ((!‘(𝑁𝑘)) · (!‘𝑘)) = ((!‘(𝑁𝐾)) · (!‘𝐾)))
1413oveq2d 6565 . . 3 (𝑘 = 𝐾 → ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
159, 14ifbieq1d 4059 . 2 (𝑘 = 𝐾 → if(𝑘 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))), 0) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0))
16 df-bc 12952 . 2 C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
17 ovex 6577 . . 3 ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))) ∈ V
18 c0ex 9913 . . 3 0 ∈ V
1917, 18ifex 4106 . 2 if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0) ∈ V
208, 15, 16, 19ovmpt2 6694 1 ((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ifcif 4036  ‘cfv 5804  (class class class)co 6549  0cc0 9815   · cmul 9820   − cmin 10145   / cdiv 10563  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  !cfa 12922  Ccbc 12951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-bc 12952 This theorem is referenced by:  bcval2  12954  bcval3  12955  bcneg1  30875  bccolsum  30878  fwddifnp1  31442
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