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Theorem bccval 38357
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 38356 . . 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 793 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑐 = 𝐶)
4 simprr 795 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 6653 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
64fveq2d 6182 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
75, 6oveq12d 6653 . 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 6665 . 2 (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
112, 7, 8, 9, 10ovmpt2d 6773 1 (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cfv 5876  (class class class)co 6635  cmpt2 6637  cc 9919   / cdiv 10669  0cn0 11277  !cfa 13043   FallFac cfallfac 14716  C𝑐cbcc 38355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-bcc 38356
This theorem is referenced by:  bcccl  38358  bcc0  38359  bccp1k  38360  bccn0  38362  bccbc  38364  binomcxplemwb  38367
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