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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 13485 | . . 3 ⊢ (𝐾 ∈ (0...(𝑁 − 1)) → (((𝑁 − 1) + 1)C𝐾) = (((𝑁 − 1)C𝐾) · (((𝑁 − 1) + 1) / (((𝑁 − 1) + 1) − 𝐾)))) | |
2 | 1 | adantl 474 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1)C𝐾) = (((𝑁 − 1)C𝐾) · (((𝑁 − 1) + 1) / (((𝑁 − 1) + 1) − 𝐾)))) |
3 | simpl 475 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ) | |
4 | 3 | nncnd 11451 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℂ) |
5 | 1cnd 10428 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → 1 ∈ ℂ) | |
6 | 4, 5 | npcand 10796 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
7 | 6 | oveq1d 6985 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1)C𝐾) = (𝑁C𝐾)) |
8 | 6 | oveq1d 6985 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1) − 𝐾) = (𝑁 − 𝐾)) |
9 | 6, 8 | oveq12d 6988 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (((𝑁 − 1) + 1) / (((𝑁 − 1) + 1) − 𝐾)) = (𝑁 / (𝑁 − 𝐾))) |
10 | 9 | oveq2d 6986 | . 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 2816 | 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 387 = wceq 1507 ∈ wcel 2050 (class class class)co 6970 0cc0 10329 1c1 10330 + caddc 10332 · cmul 10334 − cmin 10664 / cdiv 11092 ℕcn 11433 ...cfz 12702 Ccbc 13471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-seq 13179 df-fac 13443 df-bc 13472 |
This theorem is referenced by: bcprod 32490 |
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