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Mirrors > Home > MPE Home > Th. List > Mathboxes > bcneg1 | Structured version Visualization version GIF version |
Description: The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.) |
Ref | Expression |
---|---|
bcneg1 | ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1z 11413 | . . 3 ⊢ -1 ∈ ℤ | |
2 | bcval 13091 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ -1 ∈ ℤ) → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) | |
3 | 1, 2 | mpan2 707 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) |
4 | neg1lt0 11127 | . . . . . 6 ⊢ -1 < 0 | |
5 | neg1rr 11125 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
6 | 0re 10040 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 5, 6 | ltnlei 10158 | . . . . . 6 ⊢ (-1 < 0 ↔ ¬ 0 ≤ -1) |
8 | 4, 7 | mpbi 220 | . . . . 5 ⊢ ¬ 0 ≤ -1 |
9 | 8 | intnanr 961 | . . . 4 ⊢ ¬ (0 ≤ -1 ∧ -1 ≤ 𝑁) |
10 | nn0z 11400 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
11 | 0z 11388 | . . . . . 6 ⊢ 0 ∈ ℤ | |
12 | elfz 12332 | . . . . . 6 ⊢ ((-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1 ∈ (0...𝑁) ↔ (0 ≤ -1 ∧ -1 ≤ 𝑁))) | |
13 | 1, 11, 12 | mp3an12 1414 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (-1 ∈ (0...𝑁) ↔ (0 ≤ -1 ∧ -1 ≤ 𝑁))) |
14 | 10, 13 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (-1 ∈ (0...𝑁) ↔ (0 ≤ -1 ∧ -1 ≤ 𝑁))) |
15 | 9, 14 | mtbiri 317 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ¬ -1 ∈ (0...𝑁)) |
16 | 15 | iffalsed 4097 | . 2 ⊢ (𝑁 ∈ ℕ0 → if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0) = 0) |
17 | 3, 16 | eqtrd 2656 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕ0cn0 11292 ℤcz 11377 ...cfz 12326 !cfa 13060 Ccbc 13089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-fz 12327 df-bc 13090 |
This theorem is referenced by: fwddifnp1 32272 |
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