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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 12367 | . . 3 ⊢ -1 ∈ ℤ | |
2 | bcval 14029 | . . 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 12101 | . . . . . 6 ⊢ -1 < 0 | |
5 | neg1rr 12099 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
6 | 0re 10988 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 5, 6 | ltnlei 11107 | . . . . . 6 ⊢ (-1 < 0 ↔ ¬ 0 ≤ -1) |
8 | 4, 7 | mpbi 229 | . . . . 5 ⊢ ¬ 0 ≤ -1 |
9 | 8 | intnanr 488 | . . . 4 ⊢ ¬ (0 ≤ -1 ∧ -1 ≤ 𝑁) |
10 | nn0z 12354 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
11 | 0z 12341 | . . . . . 6 ⊢ 0 ∈ ℤ | |
12 | elfz 13256 | . . . . . 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 4476 | . 2 ⊢ (𝑁 ∈ ℕ0 → if(-1 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − -1)) · (!‘-1))), 0) = 0) |
17 | 3, 16 | eqtrd 2780 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ifcif 4465 class class class wbr 5079 ‘cfv 6432 (class class class)co 7272 0cc0 10882 1c1 10883 · cmul 10887 < clt 11020 ≤ cle 11021 − cmin 11216 -cneg 11217 / cdiv 11643 ℕ0cn0 12244 ℤcz 12330 ...cfz 13250 !cfa 13998 Ccbc 14027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-n0 12245 df-z 12331 df-fz 13251 df-bc 14028 |
This theorem is referenced by: fwddifnp1 34476 |
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