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Mirrors > Home > ILE Home > Th. List > bcn1 | 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 9154 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | 1eluzge0 9550 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘0) | |
3 | 2 | a1i 9 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ (ℤ≥‘0)) |
4 | elnnuz 9540 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
5 | 4 | biimpi 120 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
6 | elfzuzb 9992 | . . . . . 6 ⊢ (1 ∈ (0...𝑁) ↔ (1 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘1))) | |
7 | 3, 5, 6 | sylanbrc 417 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ (0...𝑁)) |
8 | bcval2 10701 | . . . . 5 ⊢ (1 ∈ (0...𝑁) → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) | |
9 | 7, 8 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) |
10 | facnn2 10685 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁)) | |
11 | fac1 10680 | . . . . . . 7 ⊢ (!‘1) = 1 | |
12 | 11 | oveq2i 5879 | . . . . . 6 ⊢ ((!‘(𝑁 − 1)) · (!‘1)) = ((!‘(𝑁 − 1)) · 1) |
13 | nnm1nn0 9193 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
14 | 13 | faccld 10687 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℕ) |
15 | 14 | nncnd 8909 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℂ) |
16 | 15 | mulid1d 7952 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · 1) = (!‘(𝑁 − 1))) |
17 | 12, 16 | eqtrid 2222 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · (!‘1)) = (!‘(𝑁 − 1))) |
18 | 10, 17 | oveq12d 5886 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1))) = (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1)))) |
19 | nncn 8903 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
20 | 14 | nnap0d 8941 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) # 0) |
21 | 19, 15, 20 | divcanap3d 8728 | . . . 4 ⊢ (𝑁 ∈ ℕ → (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1))) = 𝑁) |
22 | 9, 18, 21 | 3eqtrd 2214 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = 𝑁) |
23 | 0nn0 9167 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
24 | 1z 9255 | . . . . 5 ⊢ 1 ∈ ℤ | |
25 | 0lt1 8061 | . . . . . 6 ⊢ 0 < 1 | |
26 | 25 | olci 732 | . . . . 5 ⊢ (1 < 0 ∨ 0 < 1) |
27 | bcval4 10703 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℤ ∧ (1 < 0 ∨ 0 < 1)) → (0C1) = 0) | |
28 | 23, 24, 26, 27 | mp3an 1337 | . . . 4 ⊢ (0C1) = 0 |
29 | oveq1 5875 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁C1) = (0C1)) | |
30 | eqeq12 2190 | . . . . 5 ⊢ (((𝑁C1) = (0C1) ∧ 𝑁 = 0) → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) | |
31 | 29, 30 | mpancom 422 | . . . 4 ⊢ (𝑁 = 0 → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) |
32 | 28, 31 | mpbiri 168 | . . 3 ⊢ (𝑁 = 0 → (𝑁C1) = 𝑁) |
33 | 22, 32 | jaoi 716 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁C1) = 𝑁) |
34 | 1, 33 | sylbi 121 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 0cc0 7789 1c1 7790 · cmul 7794 < clt 7969 − cmin 8105 / cdiv 8605 ℕcn 8895 ℕ0cn0 9152 ℤcz 9229 ℤ≥cuz 9504 ...cfz 9982 !cfa 10676 Ccbc 10698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-n0 9153 df-z 9230 df-uz 9505 df-q 9596 df-fz 9983 df-seqfrec 10419 df-fac 10677 df-bc 10699 |
This theorem is referenced by: bcnp1n 10710 bcn2m1 10720 bcn2p1 10721 bcnm1 10723 |
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