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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 43086 | . . 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 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑐 = 𝐶) | |
4 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾) | |
5 | 3, 4 | oveq12d 7426 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾)) |
6 | 4 | fveq2d 6895 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾)) |
7 | 5, 6 | oveq12d 7426 | . 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 7443 | . 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V) | |
11 | 2, 7, 8, 9, 10 | ovmpod 7559 | 1 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ℂcc 11107 / cdiv 11870 ℕ0cn0 12471 !cfa 14232 FallFac cfallfac 15947 C𝑐cbcc 43085 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-bcc 43086 |
This theorem is referenced by: bcccl 43088 bcc0 43089 bccp1k 43090 bccn0 43092 bccbc 43094 binomcxplemwb 43097 |
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