Theorem subfaclefac 35763

Description: The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)

Hypotheses

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)))

Assertion

Ref	Expression
subfaclefac	⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) ≤ (!‘𝑁))

Distinct variable groups:   𝑓,𝑛,𝑥,𝑦,𝑁   𝐷,𝑛   𝑆,𝑛,𝑥,𝑦

Allowed substitution hints:   𝐷(𝑥, 𝑦, 𝑓)   𝑆(𝑓)

Proof of Theorem subfaclefac

Step	Hyp	Ref	Expression
1		anidm 575	. . . . . 6 ⊢ ((𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))
2	1	abbii 2829	. . . . 5 ⊢ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))} = {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}
3		fzfid 14041	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
4		deranglem 35753	. . . . . 6 ⊢ ((1...𝑁) ∈ Fin → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))} ∈ Fin)
5	3, 4	syl 18	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))} ∈ Fin)
6	2, 5	eqeltrrid 2867	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
7		simpl 488	. . . . 5 ⊢ ((𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦) → 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))
8	7	ss2abi 4017	. . . 4 ⊢ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}
9		ssdomg 9010	. . . 4 ⊢ ({𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin → ({𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
10	6, 8, 9	mpisyl 22	. . 3 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
11		deranglem 35753	. . . . 5 ⊢ ((1...𝑁) ∈ Fin → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
12	3, 11	syl 18	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
13		hashdom 14447	. . . 4 ⊢ (({𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → ((♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
14	12, 6, 13	syl2anc 596	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
15	10, 14	mpbird 260	. 2 ⊢ (𝑁 ∈ ℕ0 → (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
16		derang.d	. . . 4 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
17		subfac.n	. . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
18	16, 17	subfacval 35760	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁)))
19	16	derangval 35754	. . . 4 ⊢ ((1...𝑁) ∈ Fin → (𝐷‘(1...𝑁)) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}))
20	3, 19	syl 18	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝐷‘(1...𝑁)) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}))
21	18, 20	eqtrd 2797	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}))
22		hashfac 14527	. . . 4 ⊢ ((1...𝑁) ∈ Fin → (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
23	3, 22	syl 18	. . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
24		hashfz1 14414	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
25	24	fveq2d 6886	. . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘(♯‘(1...𝑁))) = (!‘𝑁))
26	23, 25	eqtr2d 2798	. 2 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) = (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
27	15, 21, 26	3brtr4d 5141	1 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) ≤ (!‘𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740   ≠ wne 2957  ∀wral 3078   ⊆ wss 3902   class class class wbr 5107   ↦ cmpt 5190  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7417   ≼ cdom 8954  Fincfn 8956  1c1 11129   ≤ cle 11272  ℕ₀cn0 12532  ...cfz 13565  !cfa 14341  ♯chash 14398

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-dju 9910  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-n0 12533  df-xnn0 12606  df-z 12620  df-uz 12892  df-fz 13566  df-seq 14070  df-fac 14342  df-bc 14371  df-hash 14399

This theorem is used by: (None)