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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 41041 | . . 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 7153 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (𝑐 FallFac 𝑘) = (𝐶 FallFac 𝐾)) |
6 | 4 | fveq2d 6649 | . . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑘 = 𝐾)) → (!‘𝑘) = (!‘𝐾)) |
7 | 5, 6 | oveq12d 7153 | . 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 7170 | . 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ V) | |
11 | 2, 7, 8, 9, 10 | ovmpod 7281 | 1 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ℂcc 10524 / cdiv 11286 ℕ0cn0 11885 !cfa 13629 FallFac cfallfac 15350 C𝑐cbcc 41040 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-bcc 41041 |
This theorem is referenced by: bcccl 41043 bcc0 41044 bccp1k 41045 bccn0 41047 bccbc 41049 binomcxplemwb 41052 |
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