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Mirrors > Home > MPE Home > Th. List > Mathboxes > bcm1nt | Structured version Visualization version GIF version |
Description: The proportion of one bionmial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.) |
Ref | Expression |
---|---|
bcm1nt | ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcp1n 13847 | . . 3 ⊢ (𝐾 ∈ (0...(𝑁 − 1)) → (((𝑁 − 1) + 1)C𝐾) = (((𝑁 − 1)C𝐾) · (((𝑁 − 1) + 1) / (((𝑁 − 1) + 1) − 𝐾)))) | |
2 | 1 | adantl 485 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1)C𝐾) = (((𝑁 − 1)C𝐾) · (((𝑁 − 1) + 1) / (((𝑁 − 1) + 1) − 𝐾)))) |
3 | simpl 486 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ) | |
4 | 3 | nncnd 11811 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℂ) |
5 | 1cnd 10793 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → 1 ∈ ℂ) | |
6 | 4, 5 | npcand 11158 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
7 | 6 | oveq1d 7206 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1)C𝐾) = (𝑁C𝐾)) |
8 | 6 | oveq1d 7206 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1) − 𝐾) = (𝑁 − 𝐾)) |
9 | 6, 8 | oveq12d 7209 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1) / (((𝑁 − 1) + 1) − 𝐾)) = (𝑁 / (𝑁 − 𝐾))) |
10 | 9 | oveq2d 7207 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1)C𝐾) · (((𝑁 − 1) + 1) / (((𝑁 − 1) + 1) − 𝐾))) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾)))) |
11 | 2, 7, 10 | 3eqtr3d 2779 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 − cmin 11027 / cdiv 11454 ℕcn 11795 ...cfz 13060 Ccbc 13833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-seq 13540 df-fac 13805 df-bc 13834 |
This theorem is referenced by: bcprod 33373 |
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