bccval - Mathbox for Steve Rodriguez

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Theorem bccval 39493

Description: Value of the generalized binomial coefficient, 𝐶 choose 𝐾. (Contributed by Steve Rodriguez, 22-Apr-2020.)

**Hypotheses**
Ref	Expression
bccval.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
bccval.k	⊢ (𝜑 → 𝐾 ∈ ℕ₀)

**Assertion**
Ref	Expression
bccval	⊢ (𝜑 → (𝐶C_𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Proof of Theorem bccval Dummy variables 𝑘 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bcc 39492	. . 3 ⊢ C_𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ₀ ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
2	1	a1i 11	. 2 ⊢ (𝜑 → C_𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ₀ ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘))))
3		simprl 761	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑐 = 𝐶)
4		simprr 763	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 6940	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
6	4	fveq2d 6450	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
7	5, 6	oveq12d 6940	. 2 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → ((𝑐 FallFac 𝑘) / (!‘𝑘)) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
8		bccval.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
9		bccval.k	. 2 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
10		ovexd 6956	. 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
11	2, 7, 8, 9, 10	ovmpt2d 7065	1 ⊢ (𝜑 → (𝐶C_𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ℂcc 10270 / cdiv 11032 ℕ₀cn0 11642 !cfa 13378 FallFac cfallfac 15137 C_𝑐cbcc 39491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-bcc 39492

This theorem is referenced by: bcccl 39494 bcc0 39495 bccp1k 39496 bccn0 39498 bccbc 39500 binomcxplemwb 39503

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