bccval - Mathbox for Steve Rodriguez

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Theorem bccval 40690

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 40689	. . 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 769	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑐 = 𝐶)
4		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7174	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
6	4	fveq2d 6674	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
7	5, 6	oveq12d 7174	. 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 7191	. 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
11	2, 7, 8, 9, 10	ovmpod 7302	1 ⊢ (𝜑 → (𝐶C_𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ℂcc 10535 / cdiv 11297 ℕ₀cn0 11898 !cfa 13634 FallFac cfallfac 15358 C_𝑐cbcc 40688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-bcc 40689

This theorem is referenced by: bcccl 40691 bcc0 40692 bccp1k 40693 bccn0 40695 bccbc 40697 binomcxplemwb 40700

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