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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > derangen2 | Structured version Visualization version GIF version |
Description: Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.) |
Ref | Expression |
---|---|
derang.d | ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
subfac.n | ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) |
Ref | Expression |
---|---|
derangen2 | ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (𝑆‘(♯‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 14355 | . . 3 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
2 | derang.d | . . . 4 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) | |
3 | subfac.n | . . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) | |
4 | 2, 3 | subfacval 34816 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ0 → (𝑆‘(♯‘𝐴)) = (𝐷‘(1...(♯‘𝐴)))) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → (𝑆‘(♯‘𝐴)) = (𝐷‘(1...(♯‘𝐴)))) |
6 | hashfz1 14345 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘(1...(♯‘𝐴))) = (♯‘𝐴)) | |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(1...(♯‘𝐴))) = (♯‘𝐴)) |
8 | fzfid 13978 | . . . . 5 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ∈ Fin) | |
9 | hashen 14346 | . . . . 5 ⊢ (((1...(♯‘𝐴)) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(1...(♯‘𝐴))) = (♯‘𝐴) ↔ (1...(♯‘𝐴)) ≈ 𝐴)) | |
10 | 8, 9 | mpancom 686 | . . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘(1...(♯‘𝐴))) = (♯‘𝐴) ↔ (1...(♯‘𝐴)) ≈ 𝐴)) |
11 | 7, 10 | mpbid 231 | . . 3 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴) |
12 | 2 | derangen 34815 | . . 3 ⊢ (((1...(♯‘𝐴)) ≈ 𝐴 ∧ 𝐴 ∈ Fin) → (𝐷‘(1...(♯‘𝐴))) = (𝐷‘𝐴)) |
13 | 11, 12 | mpancom 686 | . 2 ⊢ (𝐴 ∈ Fin → (𝐷‘(1...(♯‘𝐴))) = (𝐷‘𝐴)) |
14 | 5, 13 | eqtr2d 2769 | 1 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (𝑆‘(♯‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2705 ≠ wne 2937 ∀wral 3058 class class class wbr 5152 ↦ cmpt 5235 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 ≈ cen 8967 Fincfn 8970 1c1 11147 ℕ0cn0 12510 ...cfz 13524 ♯chash 14329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14330 |
This theorem is referenced by: subfacp1lem3 34825 subfacp1lem5 34827 derangfmla 34833 |
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