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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 10546.) (Contributed by Mario Carneiro, 10-Mar-2014.) |
Ref | Expression |
---|---|
bcrpcl | ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcval2 10528 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) | |
2 | elfz3nn0 9926 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | faccl 10513 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ) |
5 | fznn0sub 9868 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) | |
6 | elfznn0 9925 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0) | |
7 | faccl 10513 | . . . . 5 ⊢ ((𝑁 − 𝐾) ∈ ℕ0 → (!‘(𝑁 − 𝐾)) ∈ ℕ) | |
8 | faccl 10513 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ) | |
9 | nnmulcl 8765 | . . . . 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 409 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
12 | nnrp 9480 | . . . 4 ⊢ ((!‘𝑁) ∈ ℕ → (!‘𝑁) ∈ ℝ+) | |
13 | nnrp 9480 | . . . 4 ⊢ (((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℝ+) | |
14 | rpdivcl 9496 | . . . 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 409 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) |
17 | 1, 16 | eqeltrd 2217 | 1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ‘cfv 5131 (class class class)co 5782 0cc0 7644 · cmul 7649 − cmin 7957 / cdiv 8456 ℕcn 8744 ℕ0cn0 9001 ℝ+crp 9470 ...cfz 9821 !cfa 10503 Ccbc 10525 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-fz 9822 df-seqfrec 10250 df-fac 10504 df-bc 10526 |
This theorem is referenced by: bcp1nk 10540 bcpasc 10544 bccl2 10546 |
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