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Mirrors > Home > MPE Home > Th. List > Mathboxes > subfacp1 | Structured version Visualization version GIF version |
Description: A two-term recurrence for the subfactorial. This theorem allows us to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 31762, subfac1 31763. (Contributed by Mario Carneiro, 23-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 |
---|---|
subfacp1 | ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | derang.d | . 2 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) | |
2 | subfac.n | . 2 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) | |
3 | f1oeq1 6382 | . . . 4 ⊢ (𝑔 = 𝑓 → (𝑔:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))) | |
4 | fveq2 6448 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑔‘𝑧) = (𝑔‘𝑦)) | |
5 | id 22 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) | |
6 | 4, 5 | neeq12d 3030 | . . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑔‘𝑧) ≠ 𝑧 ↔ (𝑔‘𝑦) ≠ 𝑦)) |
7 | 6 | cbvralv 3367 | . . . . 5 ⊢ (∀𝑧 ∈ (1...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑔‘𝑦) ≠ 𝑦) |
8 | fveq1 6447 | . . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑦) = (𝑓‘𝑦)) | |
9 | 8 | neeq1d 3028 | . . . . . 6 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑦) ≠ 𝑦 ↔ (𝑓‘𝑦) ≠ 𝑦)) |
10 | 9 | ralbidv 3168 | . . . . 5 ⊢ (𝑔 = 𝑓 → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑔‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)) |
11 | 7, 10 | syl5bb 275 | . . . 4 ⊢ (𝑔 = 𝑓 → (∀𝑧 ∈ (1...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)) |
12 | 3, 11 | anbi12d 624 | . . 3 ⊢ (𝑔 = 𝑓 → ((𝑔:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑧 ∈ (1...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧) ↔ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦))) |
13 | 12 | cbvabv 2914 | . 2 ⊢ {𝑔 ∣ (𝑔:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑧 ∈ (1...(𝑁 + 1))(𝑔‘𝑧) ≠ 𝑧)} = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} |
14 | 1, 2, 13 | subfacp1lem6 31770 | 1 ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {cab 2763 ≠ wne 2969 ∀wral 3090 ↦ cmpt 4967 –1-1-onto→wf1o 6136 ‘cfv 6137 (class class class)co 6924 Fincfn 8243 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 ℕcn 11378 ℕ0cn0 11646 ...cfz 12647 ♯chash 13439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-n0 11647 df-xnn0 11719 df-z 11733 df-uz 11997 df-fz 12648 df-hash 13440 |
This theorem is referenced by: subfacval2 31772 |
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