bccval - Mathbox for Steve Rodriguez

	Mathbox for Steve Rodriguez		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > bccval		Structured version Visualization version GIF version

Theorem bccval 44615

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 44614	. . 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 771	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑐 = 𝐶)
4		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
5	3, 4	oveq12d 7378	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾))
6	4	fveq2d 6839	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾))
7	5, 6	oveq12d 7378	. 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 7395	. 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V)
11	2, 7, 8, 9, 10	ovmpod 7512	1 ⊢ (𝜑 → (𝐶C_𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3441 ‘cfv 6493 (class class class)co 7360 ∈ cmpo 7362 ℂcc 11028 / cdiv 11798 ℕ₀cn0 12405 !cfa 14200 FallFac cfallfac 15931 C_𝑐cbcc 44613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pr 5378

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-dif 3905 df-un 3907 df-ss 3919 df-nul 4287 df-if 4481 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-iota 6449 df-fun 6495 df-fv 6501 df-ov 7363 df-oprab 7364 df-mpo 7365 df-bcc 44614

This theorem is referenced by: bcccl 44616 bcc0 44617 bccp1k 44618 bccn0 44620 bccbc 44622 binomcxplemwb 44625

W3C validator