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Mirrors > Home > MPE Home > Th. List > bcn0 | Structured version Visualization version GIF version |
Description: 𝑁 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Ref | Expression |
---|---|
bcn0 | ⊢ (𝑁 ∈ ℕ0 → (𝑁C0) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elfz 12687 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
2 | bcval2 13341 | . . 3 ⊢ (0 ∈ (0...𝑁) → (𝑁C0) = ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0)))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁C0) = ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0)))) |
4 | nn0cn 11587 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
5 | 4 | subid1d 10671 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 0) = 𝑁) |
6 | 5 | fveq2d 6413 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 − 0)) = (!‘𝑁)) |
7 | fac0 13312 | . . . . . 6 ⊢ (!‘0) = 1 | |
8 | oveq12 6885 | . . . . . 6 ⊢ (((!‘(𝑁 − 0)) = (!‘𝑁) ∧ (!‘0) = 1) → ((!‘(𝑁 − 0)) · (!‘0)) = ((!‘𝑁) · 1)) | |
9 | 6, 7, 8 | sylancl 581 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((!‘(𝑁 − 0)) · (!‘0)) = ((!‘𝑁) · 1)) |
10 | faccl 13319 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
11 | 10 | nncnd 11328 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ) |
12 | 11 | mulid1d 10344 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) · 1) = (!‘𝑁)) |
13 | 9, 12 | eqtrd 2831 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((!‘(𝑁 − 0)) · (!‘0)) = (!‘𝑁)) |
14 | 13 | oveq2d 6892 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0))) = ((!‘𝑁) / (!‘𝑁))) |
15 | facne0 13322 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0) | |
16 | 11, 15 | dividd 11089 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / (!‘𝑁)) = 1) |
17 | 14, 16 | eqtrd 2831 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0))) = 1) |
18 | 3, 17 | eqtrd 2831 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C0) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 0cc0 10222 1c1 10223 · cmul 10227 − cmin 10554 / cdiv 10974 ℕ0cn0 11576 ...cfz 12576 !cfa 13309 Ccbc 13338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-seq 13052 df-fac 13310 df-bc 13339 |
This theorem is referenced by: bcnn 13348 bcpasc 13357 bccl 13358 hashbc 13482 hashf1 13486 binom 14897 bcxmas 14902 bpoly1 15115 bpoly2 15121 bpoly3 15122 bpoly4 15123 sylow1lem1 18323 srgbinom 18858 bclbnd 25354 dvnmul 40890 |
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