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Theorem bcval 13384
 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 13385 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 6913 . . . 4 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
21eleq2d 2892 . . 3 (𝑛 = 𝑁 → (𝑘 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑁)))
3 fveq2 6433 . . . 4 (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
4 fvoveq1 6928 . . . . 5 (𝑛 = 𝑁 → (!‘(𝑛𝑘)) = (!‘(𝑁𝑘)))
54oveq1d 6920 . . . 4 (𝑛 = 𝑁 → ((!‘(𝑛𝑘)) · (!‘𝑘)) = ((!‘(𝑁𝑘)) · (!‘𝑘)))
63, 5oveq12d 6923 . . 3 (𝑛 = 𝑁 → ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))) = ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))))
72, 6ifbieq1d 4329 . 2 (𝑛 = 𝑁 → if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0) = if(𝑘 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))), 0))
8 eleq1 2894 . . 3 (𝑘 = 𝐾 → (𝑘 ∈ (0...𝑁) ↔ 𝐾 ∈ (0...𝑁)))
9 oveq2 6913 . . . . . 6 (𝑘 = 𝐾 → (𝑁𝑘) = (𝑁𝐾))
109fveq2d 6437 . . . . 5 (𝑘 = 𝐾 → (!‘(𝑁𝑘)) = (!‘(𝑁𝐾)))
11 fveq2 6433 . . . . 5 (𝑘 = 𝐾 → (!‘𝑘) = (!‘𝐾))
1210, 11oveq12d 6923 . . . 4 (𝑘 = 𝐾 → ((!‘(𝑁𝑘)) · (!‘𝑘)) = ((!‘(𝑁𝐾)) · (!‘𝐾)))
1312oveq2d 6921 . . 3 (𝑘 = 𝐾 → ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
148, 13ifbieq1d 4329 . 2 (𝑘 = 𝐾 → if(𝑘 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝑘)) · (!‘𝑘))), 0) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0))
15 df-bc 13383 . 2 C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
16 ovex 6937 . . 3 ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))) ∈ V
17 c0ex 10350 . . 3 0 ∈ V
1816, 17ifex 4354 . 2 if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0) ∈ V
197, 14, 15, 18ovmpt2 7056 1 ((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ifcif 4306  ‘cfv 6123  (class class class)co 6905  0cc0 10252   · cmul 10257   − cmin 10585   / cdiv 11009  ℕ0cn0 11618  ℤcz 11704  ...cfz 12619  !cfa 13353  Ccbc 13382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-mulcl 10314  ax-i2m1 10320 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-bc 13383 This theorem is referenced by:  bcval2  13385  bcval3  13386  bcneg1  32164  bccolsum  32167  fwddifnp1  32811
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