![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > bcn1 | Structured version Visualization version GIF version |
Description: Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Ref | Expression |
---|---|
bcn1 | ⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11332 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | 1eluzge0 11770 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘0) | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ (ℤ≥‘0)) |
4 | elnnuz 11762 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
5 | 4 | biimpi 206 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
6 | elfzuzb 12374 | . . . . . 6 ⊢ (1 ∈ (0...𝑁) ↔ (1 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘1))) | |
7 | 3, 5, 6 | sylanbrc 699 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ (0...𝑁)) |
8 | bcval2 13132 | . . . . 5 ⊢ (1 ∈ (0...𝑁) → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) |
10 | facnn2 13109 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁)) | |
11 | fac1 13104 | . . . . . . 7 ⊢ (!‘1) = 1 | |
12 | 11 | oveq2i 6701 | . . . . . 6 ⊢ ((!‘(𝑁 − 1)) · (!‘1)) = ((!‘(𝑁 − 1)) · 1) |
13 | nnm1nn0 11372 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
14 | faccl 13110 | . . . . . . . . 9 ⊢ ((𝑁 − 1) ∈ ℕ0 → (!‘(𝑁 − 1)) ∈ ℕ) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℕ) |
16 | 15 | nncnd 11074 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℂ) |
17 | 16 | mulid1d 10095 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · 1) = (!‘(𝑁 − 1))) |
18 | 12, 17 | syl5eq 2697 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · (!‘1)) = (!‘(𝑁 − 1))) |
19 | 10, 18 | oveq12d 6708 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1))) = (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1)))) |
20 | nncn 11066 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
21 | 15 | nnne0d 11103 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ≠ 0) |
22 | 20, 16, 21 | divcan3d 10844 | . . . 4 ⊢ (𝑁 ∈ ℕ → (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1))) = 𝑁) |
23 | 9, 19, 22 | 3eqtrd 2689 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = 𝑁) |
24 | 0nn0 11345 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
25 | 1z 11445 | . . . . 5 ⊢ 1 ∈ ℤ | |
26 | 0lt1 10588 | . . . . . 6 ⊢ 0 < 1 | |
27 | 26 | olci 405 | . . . . 5 ⊢ (1 < 0 ∨ 0 < 1) |
28 | bcval4 13134 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℤ ∧ (1 < 0 ∨ 0 < 1)) → (0C1) = 0) | |
29 | 24, 25, 27, 28 | mp3an 1464 | . . . 4 ⊢ (0C1) = 0 |
30 | oveq1 6697 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁C1) = (0C1)) | |
31 | eqeq12 2664 | . . . . 5 ⊢ (((𝑁C1) = (0C1) ∧ 𝑁 = 0) → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) | |
32 | 30, 31 | mpancom 704 | . . . 4 ⊢ (𝑁 = 0 → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) |
33 | 29, 32 | mpbiri 248 | . . 3 ⊢ (𝑁 = 0 → (𝑁C1) = 𝑁) |
34 | 23, 33 | jaoi 393 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁C1) = 𝑁) |
35 | 1, 34 | sylbi 207 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 0cc0 9974 1c1 9975 · cmul 9979 < clt 10112 − cmin 10304 / cdiv 10722 ℕcn 11058 ℕ0cn0 11330 ℤcz 11415 ℤ≥cuz 11725 ...cfz 12364 !cfa 13100 Ccbc 13129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-seq 12842 df-fac 13101 df-bc 13130 |
This theorem is referenced by: bcnp1n 13141 bcn2m1 13151 bcn2p1 13152 bcnm1 13154 bpoly2 14832 bpoly3 14833 bpoly4 14834 jm2.23 37880 |
Copyright terms: Public domain | W3C validator |