Theorem subfaclefac 35490

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 572	. . . . . 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 2828	. . . . 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 13983	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
4		deranglem 35480	. . . . . 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 17	. . . . 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 2866	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
7		simpl 486	. . . . 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 4019	. . . 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 8977	. . . 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 21	. . 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 35480	. . . . 5 ⊢ ((1...𝑁) ∈ Fin → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
12	3, 11	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
13		hashdom 14389	. . . 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 593	. . 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 259	. 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 35487	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁)))
19	16	derangval 35481	. . . 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 17	. . 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 2796	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}))
22		hashfac 14468	. . . 4 ⊢ ((1...𝑁) ∈ Fin → (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
23	3, 22	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
24		hashfz1 14356	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
25	24	fveq2d 6867	. . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘(♯‘(1...𝑁))) = (!‘𝑁))
26	23, 25	eqtr2d 2797	. 2 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) = (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
27	15, 21, 26	3brtr4d 5131	1 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) ≤ (!‘𝑁))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075   ⊆ wss 3904   class class class wbr 5099   ↦ cmpt 5180  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392   ≼ cdom 8921  Fincfn 8923  1c1 11071   ≤ cle 11214  ℕ₀cn0 12478  ...cfz 13509  !cfa 14283  ♯chash 14340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-oadd 8436  df-er 8673  df-map 8805  df-pm 8806  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-n0 12479  df-xnn0 12552  df-z 12566  df-uz 12837  df-fz 13510  df-seq 14012  df-fac 14284  df-bc 14313  df-hash 14341

This theorem is referenced by: (None)