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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 12356 | . . 3 ⊢ -1 ∈ ℤ | |
2 | bcval 14018 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ -1 ∈ ℤ) → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) | |
3 | 1, 2 | mpan2 688 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) |
4 | neg1lt0 12090 | . . . . . 6 ⊢ -1 < 0 | |
5 | neg1rr 12088 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
6 | 0re 10977 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 5, 6 | ltnlei 11096 | . . . . . 6 ⊢ (-1 < 0 ↔ ¬ 0 ≤ -1) |
8 | 4, 7 | mpbi 229 | . . . . 5 ⊢ ¬ 0 ≤ -1 |
9 | 8 | intnanr 488 | . . . 4 ⊢ ¬ (0 ≤ -1 ∧ -1 ≤ 𝑁) |
10 | nn0z 12343 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
11 | 0z 12330 | . . . . . 6 ⊢ 0 ∈ ℤ | |
12 | elfz 13245 | . . . . . 6 ⊢ ((-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1 ∈ (0...𝑁) ↔ (0 ≤ -1 ∧ -1 ≤ 𝑁))) | |
13 | 1, 11, 12 | mp3an12 1450 | . . . . 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 327 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ¬ -1 ∈ (0...𝑁)) |
16 | 15 | iffalsed 4470 | . 2 ⊢ (𝑁 ∈ ℕ0 → if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0) = 0) |
17 | 3, 16 | eqtrd 2778 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ifcif 4459 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕ0cn0 12233 ℤcz 12319 ...cfz 13239 !cfa 13987 Ccbc 14016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-fz 13240 df-bc 14017 |
This theorem is referenced by: fwddifnp1 34467 |
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