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Mirrors > Home > ILE Home > Th. List > permnn | GIF version |
Description: The number of permutations of 𝑁 − 𝑅 objects from a collection of 𝑁 objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.) |
Ref | Expression |
---|---|
permnn | ⊢ (𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznn0 9887 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → 𝑅 ∈ ℕ0) | |
2 | 1 | faccld 10475 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑅) ∈ ℕ) |
3 | fznn0sub 9830 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (𝑁 − 𝑅) ∈ ℕ0) | |
4 | 3 | faccld 10475 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (!‘(𝑁 − 𝑅)) ∈ ℕ) |
5 | 4, 2 | nnmulcld 8762 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → ((!‘(𝑁 − 𝑅)) · (!‘𝑅)) ∈ ℕ) |
6 | elfz3nn0 9888 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
7 | faccl 10474 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
8 | 7 | nncnd 8727 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ) |
9 | 6, 8 | syl 14 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑁) ∈ ℂ) |
10 | 4 | nncnd 8727 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (!‘(𝑁 − 𝑅)) ∈ ℂ) |
11 | 2 | nncnd 8727 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑅) ∈ ℂ) |
12 | 2 | nnap0d 8759 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑅) # 0) |
13 | 10, 11, 12 | divcanap4d 8549 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (((!‘(𝑁 − 𝑅)) · (!‘𝑅)) / (!‘𝑅)) = (!‘(𝑁 − 𝑅))) |
14 | 13, 4 | eqeltrd 2214 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → (((!‘(𝑁 − 𝑅)) · (!‘𝑅)) / (!‘𝑅)) ∈ ℕ) |
15 | bcval2 10489 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (𝑁C𝑅) = ((!‘𝑁) / ((!‘(𝑁 − 𝑅)) · (!‘𝑅)))) | |
16 | bccl2 10507 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (𝑁C𝑅) ∈ ℕ) | |
17 | 15, 16 | eqeltrrd 2215 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → ((!‘𝑁) / ((!‘(𝑁 − 𝑅)) · (!‘𝑅))) ∈ ℕ) |
18 | nndivtr 8755 | . 2 ⊢ ((((!‘𝑅) ∈ ℕ ∧ ((!‘(𝑁 − 𝑅)) · (!‘𝑅)) ∈ ℕ ∧ (!‘𝑁) ∈ ℂ) ∧ ((((!‘(𝑁 − 𝑅)) · (!‘𝑅)) / (!‘𝑅)) ∈ ℕ ∧ ((!‘𝑁) / ((!‘(𝑁 − 𝑅)) · (!‘𝑅))) ∈ ℕ)) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ) | |
19 | 2, 5, 9, 14, 17, 18 | syl32anc 1224 | 1 ⊢ (𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 0cc0 7613 · cmul 7618 − cmin 7926 / cdiv 8425 ℕcn 8713 ℕ0cn0 8970 ...cfz 9783 !cfa 10464 Ccbc 10486 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-seqfrec 10212 df-fac 10465 df-bc 10487 |
This theorem is referenced by: eirraplem 11472 |
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