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Theorem bccval 37559
Description: Value of the generalized binomial coefficient, 𝐶 choose 𝐾. (Contributed by Steve Rodriguez, 22-Apr-2020.)
Hypotheses
Ref Expression
bccval.c (𝜑𝐶 ∈ ℂ)
bccval.k (𝜑𝐾 ∈ ℕ0)
Assertion
Ref Expression
bccval (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Proof of Theorem bccval
Dummy variables 𝑘 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-bcc 37558 . . 3 C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
21a1i 11 . 2 (𝜑 → C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘))))
3 simprl 790 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑐 = 𝐶)
4 simprr 792 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 6567 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
64fveq2d 6107 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
75, 6oveq12d 6567 . 2 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → ((𝑐 FallFac 𝑘) / (!‘𝑘)) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
8 bccval.c . 2 (𝜑𝐶 ∈ ℂ)
9 bccval.k . 2 (𝜑𝐾 ∈ ℕ0)
10 ovex 6577 . . 3 ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V
1110a1i 11 . 2 (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
122, 7, 8, 9, 11ovmpt2d 6686 1 (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cfv 5804  (class class class)co 6549  cmpt2 6551  cc 9813   / cdiv 10563  0cn0 11169  !cfa 12922   FallFac cfallfac 14574  C𝑐cbcc 37557
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
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-bcc 37558
This theorem is referenced by:  bcccl  37560  bcc0  37561  bccp1k  37562  bccn0  37564  bccbc  37566  binomcxplemwb  37569
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