bccval - Mathbox for Steve Rodriguez

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Theorem bccval 44299

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 44298	. . 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 770	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑐 = 𝐶)
4		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7412	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
6	4	fveq2d 6869	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
7	5, 6	oveq12d 7412	. 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 7429	. 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
11	2, 7, 8, 9, 10	ovmpod 7548	1 ⊢ (𝜑 → (𝐶C_𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3455 ‘cfv 6519 (class class class)co 7394 ∈ cmpo 7396 ℂcc 11084 / cdiv 11851 ℕ₀cn0 12458 !cfa 14248 FallFac cfallfac 15977 C_𝑐cbcc 44297

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5259 ax-nul 5269 ax-pr 5395

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-ral 3047 df-rex 3056 df-rab 3412 df-v 3457 df-sbc 3762 df-dif 3925 df-un 3927 df-ss 3939 df-nul 4305 df-if 4497 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-br 5116 df-opab 5178 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-iota 6472 df-fun 6521 df-fv 6527 df-ov 7397 df-oprab 7398 df-mpo 7399 df-bcc 44298

This theorem is referenced by: bcccl 44300 bcc0 44301 bccp1k 44302 bccn0 44304 bccbc 44306 binomcxplemwb 44309

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