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Theorem bccval 42710
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 42709 . . 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 770 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑐 = 𝐶)
4 simprr 772 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → 𝑘 = 𝐾)
53, 4oveq12d 7379 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
64fveq2d 6850 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
75, 6oveq12d 7379 . 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 7396 . 2 (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
112, 7, 8, 9, 10ovmpod 7511 1 (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cfv 6500  (class class class)co 7361  cmpo 7363  cc 11057   / cdiv 11820  0cn0 12421  !cfa 14182   FallFac cfallfac 15895  C𝑐cbcc 42708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-bcc 42709
This theorem is referenced by:  bcccl  42711  bcc0  42712  bccp1k  42713  bccn0  42715  bccbc  42717  binomcxplemwb  42720
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