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Mirrors > Home > MPE Home > Th. List > bcval | Structured version Visualization version GIF version |
Description: Value of the binomial coefficient, 𝑁 choose 𝐾. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝐾 ≤ 𝑁 does not hold. See bcval2 13132 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Ref | Expression |
---|---|
bcval | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6698 | . . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) | |
2 | 1 | eleq2d 2716 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑘 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑁))) |
3 | fveq2 6229 | . . . 4 ⊢ (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁)) | |
4 | oveq1 6697 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘)) | |
5 | 4 | fveq2d 6233 | . . . . 5 ⊢ (𝑛 = 𝑁 → (!‘(𝑛 − 𝑘)) = (!‘(𝑁 − 𝑘))) |
6 | 5 | oveq1d 6705 | . . . 4 ⊢ (𝑛 = 𝑁 → ((!‘(𝑛 − 𝑘)) · (!‘𝑘)) = ((!‘(𝑁 − 𝑘)) · (!‘𝑘))) |
7 | 3, 6 | oveq12d 6708 | . . 3 ⊢ (𝑛 = 𝑁 → ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))) = ((!‘𝑁) / ((!‘(𝑁 − 𝑘)) · (!‘𝑘)))) |
8 | 2, 7 | ifbieq1d 4142 | . 2 ⊢ (𝑛 = 𝑁 → if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0) = if(𝑘 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝑘)) · (!‘𝑘))), 0)) |
9 | eleq1 2718 | . . 3 ⊢ (𝑘 = 𝐾 → (𝑘 ∈ (0...𝑁) ↔ 𝐾 ∈ (0...𝑁))) | |
10 | oveq2 6698 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑁 − 𝑘) = (𝑁 − 𝐾)) | |
11 | 10 | fveq2d 6233 | . . . . 5 ⊢ (𝑘 = 𝐾 → (!‘(𝑁 − 𝑘)) = (!‘(𝑁 − 𝐾))) |
12 | fveq2 6229 | . . . . 5 ⊢ (𝑘 = 𝐾 → (!‘𝑘) = (!‘𝐾)) | |
13 | 11, 12 | oveq12d 6708 | . . . 4 ⊢ (𝑘 = 𝐾 → ((!‘(𝑁 − 𝑘)) · (!‘𝑘)) = ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) |
14 | 13 | oveq2d 6706 | . . 3 ⊢ (𝑘 = 𝐾 → ((!‘𝑁) / ((!‘(𝑁 − 𝑘)) · (!‘𝑘))) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) |
15 | 9, 14 | ifbieq1d 4142 | . 2 ⊢ (𝑘 = 𝐾 → if(𝑘 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝑘)) · (!‘𝑘))), 0) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0)) |
16 | df-bc 13130 | . 2 ⊢ C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)) | |
17 | ovex 6718 | . . 3 ⊢ ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ V | |
18 | c0ex 10072 | . . 3 ⊢ 0 ∈ V | |
19 | 17, 18 | ifex 4189 | . 2 ⊢ if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0) ∈ V |
20 | 8, 15, 16, 19 | ovmpt2 6838 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ifcif 4119 ‘cfv 5926 (class class class)co 6690 0cc0 9974 · cmul 9979 − cmin 10304 / cdiv 10722 ℕ0cn0 11330 ℤcz 11415 ...cfz 12364 !cfa 13100 Ccbc 13129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-mulcl 10036 ax-i2m1 10042 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-bc 13130 |
This theorem is referenced by: bcval2 13132 bcval3 13133 bcneg1 31748 bccolsum 31751 fwddifnp1 32397 |
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