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 11294 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | 1eluzge0 11732 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘0) | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ (ℤ≥‘0)) |
4 | elnnuz 11724 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
5 | 4 | biimpi 206 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
6 | elfzuzb 12336 | . . . . . 6 ⊢ (1 ∈ (0...𝑁) ↔ (1 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘1))) | |
7 | 3, 5, 6 | sylanbrc 698 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ (0...𝑁)) |
8 | bcval2 13092 | . . . . 5 ⊢ (1 ∈ (0...𝑁) → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) |
10 | facnn2 13069 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁)) | |
11 | fac1 13064 | . . . . . . 7 ⊢ (!‘1) = 1 | |
12 | 11 | oveq2i 6661 | . . . . . 6 ⊢ ((!‘(𝑁 − 1)) · (!‘1)) = ((!‘(𝑁 − 1)) · 1) |
13 | nnm1nn0 11334 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
14 | faccl 13070 | . . . . . . . . 9 ⊢ ((𝑁 − 1) ∈ ℕ0 → (!‘(𝑁 − 1)) ∈ ℕ) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℕ) |
16 | 15 | nncnd 11036 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℂ) |
17 | 16 | mulid1d 10057 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · 1) = (!‘(𝑁 − 1))) |
18 | 12, 17 | syl5eq 2668 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · (!‘1)) = (!‘(𝑁 − 1))) |
19 | 10, 18 | oveq12d 6668 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1))) = (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1)))) |
20 | nncn 11028 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
21 | 15 | nnne0d 11065 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ≠ 0) |
22 | 20, 16, 21 | divcan3d 10806 | . . . 4 ⊢ (𝑁 ∈ ℕ → (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1))) = 𝑁) |
23 | 9, 19, 22 | 3eqtrd 2660 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = 𝑁) |
24 | 0nn0 11307 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
25 | 1z 11407 | . . . . 5 ⊢ 1 ∈ ℤ | |
26 | 0lt1 10550 | . . . . . 6 ⊢ 0 < 1 | |
27 | 26 | olci 406 | . . . . 5 ⊢ (1 < 0 ∨ 0 < 1) |
28 | bcval4 13094 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℤ ∧ (1 < 0 ∨ 0 < 1)) → (0C1) = 0) | |
29 | 24, 25, 27, 28 | mp3an 1424 | . . . 4 ⊢ (0C1) = 0 |
30 | oveq1 6657 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁C1) = (0C1)) | |
31 | eqeq12 2635 | . . . . 5 ⊢ (((𝑁C1) = (0C1) ∧ 𝑁 = 0) → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) | |
32 | 30, 31 | mpancom 703 | . . . 4 ⊢ (𝑁 = 0 → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) |
33 | 29, 32 | mpbiri 248 | . . 3 ⊢ (𝑁 = 0 → (𝑁C1) = 𝑁) |
34 | 23, 33 | jaoi 394 | . 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 383 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ0cn0 11292 ℤcz 11377 ℤ≥cuz 11687 ...cfz 12326 !cfa 13060 Ccbc 13089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-fac 13061 df-bc 13090 |
This theorem is referenced by: bcnp1n 13101 bcn2m1 13111 bcn2p1 13112 bcnm1 13114 bpoly2 14788 bpoly3 14789 bpoly4 14790 jm2.23 37563 |
Copyright terms: Public domain | W3C validator |