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Mirrors > Home > MPE Home > Th. List > Mathboxes > bcneg1 | Structured version Visualization version GIF version |
Description: The binomial coefficient 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 12650 | . . 3 ⊢ -1 ∈ ℤ | |
2 | bcval 14321 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ -1 ∈ ℤ) → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) |
4 | neg1lt0 12381 | . . . . . 6 ⊢ -1 < 0 | |
5 | neg1rr 12379 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
6 | 0re 11266 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 5, 6 | ltnlei 11385 | . . . . . 6 ⊢ (-1 < 0 ↔ ¬ 0 ≤ -1) |
8 | 4, 7 | mpbi 229 | . . . . 5 ⊢ ¬ 0 ≤ -1 |
9 | 8 | intnanr 486 | . . . 4 ⊢ ¬ (0 ≤ -1 ∧ -1 ≤ 𝑁) |
10 | nn0z 12635 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
11 | 0z 12621 | . . . . . 6 ⊢ 0 ∈ ℤ | |
12 | elfz 13544 | . . . . . 6 ⊢ ((-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1 ∈ (0...𝑁) ↔ (0 ≤ -1 ∧ -1 ≤ 𝑁))) | |
13 | 1, 11, 12 | mp3an12 1448 | . . . . 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 326 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ¬ -1 ∈ (0...𝑁)) |
16 | 15 | iffalsed 4544 | . 2 ⊢ (𝑁 ∈ ℕ0 → if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0) = 0) |
17 | 3, 16 | eqtrd 2766 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ifcif 4533 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 0cc0 11158 1c1 11159 · cmul 11163 < clt 11298 ≤ cle 11299 − cmin 11494 -cneg 11495 / cdiv 11921 ℕ0cn0 12524 ℤcz 12610 ...cfz 13538 !cfa 14290 Ccbc 14319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-fz 13539 df-bc 14320 |
This theorem is referenced by: fwddifnp1 35989 |
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