![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > permnn | Structured version Visualization version 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 13627 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → 𝑅 ∈ ℕ0) | |
2 | 1 | faccld 14276 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑅) ∈ ℕ) |
3 | fznn0sub 13566 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (𝑁 − 𝑅) ∈ ℕ0) | |
4 | 3 | faccld 14276 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (!‘(𝑁 − 𝑅)) ∈ ℕ) |
5 | 4, 2 | nnmulcld 12296 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → ((!‘(𝑁 − 𝑅)) · (!‘𝑅)) ∈ ℕ) |
6 | elfz3nn0 13628 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
7 | faccl 14275 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
8 | 7 | nncnd 12259 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ) |
9 | 6, 8 | syl 17 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑁) ∈ ℂ) |
10 | 4 | nncnd 12259 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (!‘(𝑁 − 𝑅)) ∈ ℂ) |
11 | 2 | nncnd 12259 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑅) ∈ ℂ) |
12 | facne0 14278 | . . . . 5 ⊢ (𝑅 ∈ ℕ0 → (!‘𝑅) ≠ 0) | |
13 | 1, 12 | syl 17 | . . . 4 ⊢ (𝑅 ∈ (0...𝑁) → (!‘𝑅) ≠ 0) |
14 | 10, 11, 13 | divcan4d 12027 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (((!‘(𝑁 − 𝑅)) · (!‘𝑅)) / (!‘𝑅)) = (!‘(𝑁 − 𝑅))) |
15 | 14, 4 | eqeltrd 2829 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → (((!‘(𝑁 − 𝑅)) · (!‘𝑅)) / (!‘𝑅)) ∈ ℕ) |
16 | bcval2 14297 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (𝑁C𝑅) = ((!‘𝑁) / ((!‘(𝑁 − 𝑅)) · (!‘𝑅)))) | |
17 | bccl2 14315 | . . 3 ⊢ (𝑅 ∈ (0...𝑁) → (𝑁C𝑅) ∈ ℕ) | |
18 | 16, 17 | eqeltrrd 2830 | . 2 ⊢ (𝑅 ∈ (0...𝑁) → ((!‘𝑁) / ((!‘(𝑁 − 𝑅)) · (!‘𝑅))) ∈ ℕ) |
19 | nndivtr 12290 | . 2 ⊢ ((((!‘𝑅) ∈ ℕ ∧ ((!‘(𝑁 − 𝑅)) · (!‘𝑅)) ∈ ℕ ∧ (!‘𝑁) ∈ ℂ) ∧ ((((!‘(𝑁 − 𝑅)) · (!‘𝑅)) / (!‘𝑅)) ∈ ℕ ∧ ((!‘𝑁) / ((!‘(𝑁 − 𝑅)) · (!‘𝑅))) ∈ ℕ)) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ) | |
20 | 2, 5, 9, 15, 18, 19 | syl32anc 1376 | 1 ⊢ (𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2937 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 0cc0 11139 · cmul 11144 − cmin 11475 / cdiv 11902 ℕcn 12243 ℕ0cn0 12503 ...cfz 13517 !cfa 14265 Ccbc 14294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-fz 13518 df-seq 14000 df-fac 14266 df-bc 14295 |
This theorem is referenced by: eirrlem 16181 etransclem3 45625 etransclem7 45629 etransclem10 45632 etransclem24 45646 etransclem27 45649 |
Copyright terms: Public domain | W3C validator |