Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
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 13720 | . . 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 32424 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ0 → (𝑆‘(♯‘𝐴)) = (𝐷‘(1...(♯‘𝐴)))) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → (𝑆‘(♯‘𝐴)) = (𝐷‘(1...(♯‘𝐴)))) |
6 | hashfz1 13709 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘(1...(♯‘𝐴))) = (♯‘𝐴)) | |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(1...(♯‘𝐴))) = (♯‘𝐴)) |
8 | fzfid 13344 | . . . . 5 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ∈ Fin) | |
9 | hashen 13710 | . . . . 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 234 | . . 3 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴) |
12 | 2 | derangen 32423 | . . 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 2860 | 1 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (𝑆‘(♯‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {cab 2802 ≠ wne 3019 ∀wral 3141 class class class wbr 5069 ↦ cmpt 5149 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 ≈ cen 8509 Fincfn 8512 1c1 10541 ℕ0cn0 11900 ...cfz 12895 ♯chash 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: subfacp1lem3 32433 subfacp1lem5 32435 derangfmla 32441 |
Copyright terms: Public domain | W3C validator |