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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 12006 | . . 3 ⊢ -1 ∈ ℤ | |
2 | bcval 13652 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ -1 ∈ ℤ) → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) | |
3 | 1, 2 | mpan2 687 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0)) |
4 | neg1lt0 11742 | . . . . . 6 ⊢ -1 < 0 | |
5 | neg1rr 11740 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
6 | 0re 10631 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 5, 6 | ltnlei 10749 | . . . . . 6 ⊢ (-1 < 0 ↔ ¬ 0 ≤ -1) |
8 | 4, 7 | mpbi 231 | . . . . 5 ⊢ ¬ 0 ≤ -1 |
9 | 8 | intnanr 488 | . . . 4 ⊢ ¬ (0 ≤ -1 ∧ -1 ≤ 𝑁) |
10 | nn0z 11993 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
11 | 0z 11980 | . . . . . 6 ⊢ 0 ∈ ℤ | |
12 | elfz 12886 | . . . . . 6 ⊢ ((-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1 ∈ (0...𝑁) ↔ (0 ≤ -1 ∧ -1 ≤ 𝑁))) | |
13 | 1, 11, 12 | mp3an12 1442 | . . . . 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 328 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ¬ -1 ∈ (0...𝑁)) |
16 | 15 | iffalsed 4474 | . 2 ⊢ (𝑁 ∈ ℕ0 → if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0) = 0) |
17 | 3, 16 | eqtrd 2853 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ifcif 4463 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 · cmul 10530 < clt 10663 ≤ cle 10664 − cmin 10858 -cneg 10859 / cdiv 11285 ℕ0cn0 11885 ℤcz 11969 ...cfz 12880 !cfa 13621 Ccbc 13650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-fz 12881 df-bc 13651 |
This theorem is referenced by: fwddifnp1 33523 |
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