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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 9695.) (Contributed by Mario Carneiro, 10-Mar-2014.) |
Ref | Expression |
---|---|
bcrpcl | ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcval2 9677 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) | |
2 | elfz3nn0 9131 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | faccl 9662 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ) |
5 | fznn0sub 9075 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) | |
6 | elfznn0 9130 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0) | |
7 | faccl 9662 | . . . . 5 ⊢ ((𝑁 − 𝐾) ∈ ℕ0 → (!‘(𝑁 − 𝐾)) ∈ ℕ) | |
8 | faccl 9662 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ) | |
9 | nnmulcl 8060 | . . . . 5 ⊢ (((!‘(𝑁 − 𝐾)) ∈ ℕ ∧ (!‘𝐾) ∈ ℕ) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) | |
10 | 7, 8, 9 | syl2an 283 | . . . 4 ⊢ (((𝑁 − 𝐾) ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
11 | 5, 6, 10 | syl2anc 403 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
12 | nnrp 8743 | . . . 4 ⊢ ((!‘𝑁) ∈ ℕ → (!‘𝑁) ∈ ℝ+) | |
13 | nnrp 8743 | . . . 4 ⊢ (((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℝ+) | |
14 | rpdivcl 8759 | . . . 4 ⊢ (((!‘𝑁) ∈ ℝ+ ∧ ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℝ+) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) | |
15 | 12, 13, 14 | syl2an 283 | . . 3 ⊢ (((!‘𝑁) ∈ ℕ ∧ ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) |
16 | 4, 11, 15 | syl2anc 403 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) |
17 | 1, 16 | eqeltrd 2155 | 1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 ‘cfv 4922 (class class class)co 5532 0cc0 6981 · cmul 6986 − cmin 7279 / cdiv 7760 ℕcn 8039 ℕ0cn0 8288 ℝ+crp 8734 ...cfz 9029 !cfa 9652 Ccbc 9674 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-rp 8735 df-fz 9030 df-iseq 9432 df-fac 9653 df-bc 9675 |
This theorem is referenced by: bcp1nk 9689 bcpasc 9693 bccl2 9695 |
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