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Mirrors > Home > MPE Home > Th. List > bcctr | Structured version Visualization version GIF version |
Description: Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.) |
Ref | Expression |
---|---|
bcctr | ⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzctr 12658 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...(2 · 𝑁))) | |
2 | bcval2 13295 | . . 3 ⊢ (𝑁 ∈ (0...(2 · 𝑁)) → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘((2 · 𝑁) − 𝑁)) · (!‘𝑁)))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘((2 · 𝑁) − 𝑁)) · (!‘𝑁)))) |
4 | nn0cn 11503 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
5 | 4 | 2timesd 11476 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (2 · 𝑁) = (𝑁 + 𝑁)) |
6 | 5 | oveq1d 6807 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁) − 𝑁) = ((𝑁 + 𝑁) − 𝑁)) |
7 | 4, 4 | pncand 10594 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 𝑁) − 𝑁) = 𝑁) |
8 | 6, 7 | eqtrd 2805 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁) − 𝑁) = 𝑁) |
9 | 8 | fveq2d 6336 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘((2 · 𝑁) − 𝑁)) = (!‘𝑁)) |
10 | 9 | oveq1d 6807 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((!‘((2 · 𝑁) − 𝑁)) · (!‘𝑁)) = ((!‘𝑁) · (!‘𝑁))) |
11 | 10 | oveq2d 6808 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((!‘(2 · 𝑁)) / ((!‘((2 · 𝑁) − 𝑁)) · (!‘𝑁))) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁)))) |
12 | 3, 11 | eqtrd 2805 | 1 ⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6031 (class class class)co 6792 0cc0 10137 + caddc 10140 · cmul 10142 − cmin 10467 / cdiv 10885 2c2 11271 ℕ0cn0 11493 ...cfz 12532 !cfa 13263 Ccbc 13292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-bc 13293 |
This theorem is referenced by: bposlem3 25231 |
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