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Mirrors > Home > ILE Home > Th. List > bcn0 | 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 10117 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
2 | bcval2 10729 | . . 3 ⊢ (0 ∈ (0...𝑁) → (𝑁C0) = ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0)))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁C0) = ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0)))) |
4 | nn0cn 9185 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
5 | 4 | subid1d 8256 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 0) = 𝑁) |
6 | 5 | fveq2d 5519 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 − 0)) = (!‘𝑁)) |
7 | fac0 10707 | . . . . . 6 ⊢ (!‘0) = 1 | |
8 | oveq12 5883 | . . . . . 6 ⊢ (((!‘(𝑁 − 0)) = (!‘𝑁) ∧ (!‘0) = 1) → ((!‘(𝑁 − 0)) · (!‘0)) = ((!‘𝑁) · 1)) | |
9 | 6, 7, 8 | sylancl 413 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((!‘(𝑁 − 0)) · (!‘0)) = ((!‘𝑁) · 1)) |
10 | faccl 10714 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
11 | 10 | nncnd 8932 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ) |
12 | 11 | mulridd 7973 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) · 1) = (!‘𝑁)) |
13 | 9, 12 | eqtrd 2210 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((!‘(𝑁 − 0)) · (!‘0)) = (!‘𝑁)) |
14 | 13 | oveq2d 5890 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0))) = ((!‘𝑁) / (!‘𝑁))) |
15 | 10 | nnap0d 8964 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) # 0) |
16 | 11, 15 | dividapd 8742 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / (!‘𝑁)) = 1) |
17 | 14, 16 | eqtrd 2210 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) / ((!‘(𝑁 − 0)) · (!‘0))) = 1) |
18 | 3, 17 | eqtrd 2210 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C0) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 0cc0 7810 1c1 7811 · cmul 7815 − cmin 8127 / cdiv 8628 ℕ0cn0 9175 ...cfz 10007 !cfa 10704 Ccbc 10726 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-fz 10008 df-seqfrec 10445 df-fac 10705 df-bc 10727 |
This theorem is referenced by: bcnn 10736 bcpasc 10745 bccl 10746 binom 11491 bcxmas 11496 |
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