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| Description: Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) | 
| Ref | Expression | 
|---|---|
| bcval4 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ (𝐾 < 0 ∨ 𝑁 < 𝐾)) → (𝑁C𝐾) = 0) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elfzle1 13568 | . . . . . . . . 9 ⊢ (𝐾 ∈ (0...𝑁) → 0 ≤ 𝐾) | |
| 2 | 0re 11264 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
| 3 | elfzelz 13565 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ) | |
| 4 | 3 | zred 12724 | . . . . . . . . . 10 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℝ) | 
| 5 | lenlt 11340 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (0 ≤ 𝐾 ↔ ¬ 𝐾 < 0)) | |
| 6 | 2, 4, 5 | sylancr 587 | . . . . . . . . 9 ⊢ (𝐾 ∈ (0...𝑁) → (0 ≤ 𝐾 ↔ ¬ 𝐾 < 0)) | 
| 7 | 1, 6 | mpbid 232 | . . . . . . . 8 ⊢ (𝐾 ∈ (0...𝑁) → ¬ 𝐾 < 0) | 
| 8 | 7 | adantl 481 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (0...𝑁)) → ¬ 𝐾 < 0) | 
| 9 | elfzle2 13569 | . . . . . . . . 9 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ≤ 𝑁) | |
| 10 | 9 | adantl 481 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (0...𝑁)) → 𝐾 ≤ 𝑁) | 
| 11 | nn0re 12537 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
| 12 | lenlt 11340 | . . . . . . . . 9 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐾 ≤ 𝑁 ↔ ¬ 𝑁 < 𝐾)) | |
| 13 | 4, 11, 12 | syl2anr 597 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (0...𝑁)) → (𝐾 ≤ 𝑁 ↔ ¬ 𝑁 < 𝐾)) | 
| 14 | 10, 13 | mpbid 232 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (0...𝑁)) → ¬ 𝑁 < 𝐾) | 
| 15 | ioran 985 | . . . . . . 7 ⊢ (¬ (𝐾 < 0 ∨ 𝑁 < 𝐾) ↔ (¬ 𝐾 < 0 ∧ ¬ 𝑁 < 𝐾)) | |
| 16 | 8, 14, 15 | sylanbrc 583 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (0...𝑁)) → ¬ (𝐾 < 0 ∨ 𝑁 < 𝐾)) | 
| 17 | 16 | ex 412 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) → ¬ (𝐾 < 0 ∨ 𝑁 < 𝐾))) | 
| 18 | 17 | adantr 480 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ (0...𝑁) → ¬ (𝐾 < 0 ∨ 𝑁 < 𝐾))) | 
| 19 | 18 | con2d 134 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → ((𝐾 < 0 ∨ 𝑁 < 𝐾) → ¬ 𝐾 ∈ (0...𝑁))) | 
| 20 | 19 | 3impia 1117 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ (𝐾 < 0 ∨ 𝑁 < 𝐾)) → ¬ 𝐾 ∈ (0...𝑁)) | 
| 21 | bcval3 14346 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) | |
| 22 | 20, 21 | syld3an3 1410 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ (𝐾 < 0 ∨ 𝑁 < 𝐾)) → (𝑁C𝐾) = 0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 0cc0 11156 < clt 11296 ≤ cle 11297 ℕ0cn0 12528 ℤcz 12615 ...cfz 13548 Ccbc 14342 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-i2m1 11224 ax-1ne0 11225 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-xr 11300 df-le 11302 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-bc 14343 | 
| This theorem is referenced by: bc0k 14351 bcn1 14353 bcpasc 14361 hashf1 14497 binomfallfaclem2 16077 ram0 17061 srgbinomlem3 20226 srgbinomlem4 20227 basellem2 27126 bcmono 27322 cusgrsizeindb1 29469 altgsumbcALT 48274 | 
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