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Mirrors > Home > MPE Home > Th. List > Mathboxes > bccval | Structured version Visualization version GIF version |
Description: Value of the generalized binomial coefficient, 𝐶 choose 𝐾. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
Ref | Expression |
---|---|
bccval.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
bccval.k | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
Ref | Expression |
---|---|
bccval | ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bcc 43029 | . . 3 ⊢ C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → C𝑐 = (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 7422 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾)) |
6 | 4 | fveq2d 6892 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾)) |
7 | 5, 6 | oveq12d 7422 | . 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 7439 | . 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V) | |
11 | 2, 7, 8, 9, 10 | ovmpod 7555 | 1 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 ℂcc 11104 / cdiv 11867 ℕ0cn0 12468 !cfa 14229 FallFac cfallfac 15944 C𝑐cbcc 43028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-bcc 43029 |
This theorem is referenced by: bcccl 43031 bcc0 43032 bccp1k 43033 bccn0 43035 bccbc 43037 binomcxplemwb 43040 |
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