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Mirrors > Home > ILE Home > Th. List > bcrpcl | GIF version |
Description: Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10514.) (Contributed by Mario Carneiro, 10-Mar-2014.) |
Ref | Expression |
---|---|
bcrpcl | ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcval2 10496 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) | |
2 | elfz3nn0 9895 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | faccl 10481 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ) |
5 | fznn0sub 9837 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) | |
6 | elfznn0 9894 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0) | |
7 | faccl 10481 | . . . . 5 ⊢ ((𝑁 − 𝐾) ∈ ℕ0 → (!‘(𝑁 − 𝐾)) ∈ ℕ) | |
8 | faccl 10481 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ) | |
9 | nnmulcl 8741 | . . . . 5 ⊢ (((!‘(𝑁 − 𝐾)) ∈ ℕ ∧ (!‘𝐾) ∈ ℕ) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) | |
10 | 7, 8, 9 | syl2an 287 | . . . 4 ⊢ (((𝑁 − 𝐾) ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
11 | 5, 6, 10 | syl2anc 408 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
12 | nnrp 9451 | . . . 4 ⊢ ((!‘𝑁) ∈ ℕ → (!‘𝑁) ∈ ℝ+) | |
13 | nnrp 9451 | . . . 4 ⊢ (((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℝ+) | |
14 | rpdivcl 9467 | . . . 4 ⊢ (((!‘𝑁) ∈ ℝ+ ∧ ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℝ+) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) | |
15 | 12, 13, 14 | syl2an 287 | . . 3 ⊢ (((!‘𝑁) ∈ ℕ ∧ ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) |
16 | 4, 11, 15 | syl2anc 408 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) |
17 | 1, 16 | eqeltrd 2216 | 1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ‘cfv 5123 (class class class)co 5774 0cc0 7620 · cmul 7625 − cmin 7933 / cdiv 8432 ℕcn 8720 ℕ0cn0 8977 ℝ+crp 9441 ...cfz 9790 !cfa 10471 Ccbc 10493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-seqfrec 10219 df-fac 10472 df-bc 10494 |
This theorem is referenced by: bcp1nk 10508 bcpasc 10512 bccl2 10514 |
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