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Theorem bccval 43087
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 43086 . . 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 769 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑐 = 𝐶)
4 simprr 771 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 7426 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
64fveq2d 6895 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
75, 6oveq12d 7426 . 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 ovexd 7443 . 2 (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
112, 7, 8, 9, 10ovmpod 7559 1 (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cfv 6543  (class class class)co 7408  cmpo 7410  cc 11107   / cdiv 11870  0cn0 12471  !cfa 14232   FallFac cfallfac 15947  C𝑐cbcc 43085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-bcc 43086
This theorem is referenced by:  bcccl  43088  bcc0  43089  bccp1k  43090  bccn0  43092  bccbc  43094  binomcxplemwb  43097
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