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Mirrors > Home > MPE Home > Th. List > bcrpcl | Structured version Visualization version GIF version |
Description: Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 14278.) (Contributed by Mario Carneiro, 10-Mar-2014.) |
Ref | Expression |
---|---|
bcrpcl | ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcval2 14260 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) | |
2 | elfz3nn0 13590 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | 2 | faccld 14239 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (!‘𝑁) ∈ ℕ) |
4 | fznn0sub 13528 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) | |
5 | elfznn0 13589 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0) | |
6 | faccl 14238 | . . . . 5 ⊢ ((𝑁 − 𝐾) ∈ ℕ0 → (!‘(𝑁 − 𝐾)) ∈ ℕ) | |
7 | faccl 14238 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ) | |
8 | nnmulcl 12231 | . . . . 5 ⊢ (((!‘(𝑁 − 𝐾)) ∈ ℕ ∧ (!‘𝐾) ∈ ℕ) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) | |
9 | 6, 7, 8 | syl2an 597 | . . . 4 ⊢ (((𝑁 − 𝐾) ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
10 | 4, 5, 9 | syl2anc 585 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
11 | nnrp 12980 | . . . 4 ⊢ ((!‘𝑁) ∈ ℕ → (!‘𝑁) ∈ ℝ+) | |
12 | nnrp 12980 | . . . 4 ⊢ (((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℝ+) | |
13 | rpdivcl 12994 | . . . 4 ⊢ (((!‘𝑁) ∈ ℝ+ ∧ ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℝ+) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) | |
14 | 11, 12, 13 | syl2an 597 | . . 3 ⊢ (((!‘𝑁) ∈ ℕ ∧ ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) |
15 | 3, 10, 14 | syl2anc 585 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) ∈ ℝ+) |
16 | 1, 15 | eqeltrd 2834 | 1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6539 (class class class)co 7403 0cc0 11105 · cmul 11110 − cmin 11439 / cdiv 11866 ℕcn 12207 ℕ0cn0 12467 ℝ+crp 12969 ...cfz 13479 !cfa 14228 Ccbc 14257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-n0 12468 df-z 12554 df-uz 12818 df-rp 12970 df-fz 13480 df-seq 13962 df-fac 14229 df-bc 14258 |
This theorem is referenced by: bcp1nk 14272 bcpasc 14276 bccl2 14278 bcm1n 31983 |
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