bccval - Mathbox for Steve Rodriguez

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Theorem bccval 45076

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 45075	. . 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 782	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑐 = 𝐶)
4		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7430	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
6	4	fveq2d 6885	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
7	5, 6	oveq12d 7430	. 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 7447	. 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
11	2, 7, 8, 9, 10	ovmpod 7564	1 ⊢ (𝜑 → (𝐶C_𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 Vcvv 3454 ‘cfv 6536 (class class class)co 7412 ∈ cmpo 7414 ℂcc 11104 / cdiv 11877 ℕ₀cn0 12510 !cfa 14316 FallFac cfallfac 16065 C_𝑐cbcc 45074

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pr 5403

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-iota 6492 df-fun 6538 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-bcc 45075

This theorem is used by: bcccl 45077 bcc0 45078 bccp1k 45079 bccn0 45081 bccbc 45083 binomcxplemwb 45086

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