Theorem facubnd 10679

Description: An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)

Assertion

Ref	Expression
facubnd	|- ( N e. NN0 -> ( ! ` N ) <_ ( N ^ N ) )

Proof of Theorem facubnd

Dummy variables m k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5496	. . . 4 |- ( m = 0 -> ( ! ` m ) = ( ! ` 0 ) )
2		fac0 10662	. . . 4 |- ( ! ` 0 ) = 1
3	1, 2	eqtrdi 2219	. . 3 |- ( m = 0 -> ( ! ` m ) = 1 )
4		id 19	. . . . 5 |- ( m = 0 -> m = 0 )
5	4, 4	oveq12d 5871	. . . 4 |- ( m = 0 -> ( m ^ m ) = ( 0 ^ 0 ) )
6		0exp0e1 10481	. . . 4 |- ( 0 ^ 0 ) = 1
7	5, 6	eqtrdi 2219	. . 3 |- ( m = 0 -> ( m ^ m ) = 1 )
8	3, 7	breq12d 4002	. 2 |- ( m = 0 -> ( ( ! ` m ) <_ ( m ^ m ) <-> 1 <_ 1 ) )
9		fveq2 5496	. . 3 |- ( m = k -> ( ! ` m ) = ( ! ` k ) )
10		id 19	. . . 4 |- ( m = k -> m = k )
11	10, 10	oveq12d 5871	. . 3 |- ( m = k -> ( m ^ m ) = ( k ^ k ) )
12	9, 11	breq12d 4002	. 2 |- ( m = k -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` k ) <_ ( k ^ k ) ) )
13		fveq2 5496	. . 3 |- ( m = ( k + 1 ) -> ( ! ` m ) = ( ! ` ( k + 1 ) ) )
14		id 19	. . . 4 |- ( m = ( k + 1 ) -> m = ( k + 1 ) )
15	14, 14	oveq12d 5871	. . 3 |- ( m = ( k + 1 ) -> ( m ^ m ) = ( ( k + 1 ) ^ ( k + 1 ) ) )
16	13, 15	breq12d 4002	. 2 |- ( m = ( k + 1 ) -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) ) )
17		fveq2 5496	. . 3 |- ( m = N -> ( ! ` m ) = ( ! ` N ) )
18		id 19	. . . 4 |- ( m = N -> m = N )
19	18, 18	oveq12d 5871	. . 3 |- ( m = N -> ( m ^ m ) = ( N ^ N ) )
20	17, 19	breq12d 4002	. 2 |- ( m = N -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` N ) <_ ( N ^ N ) ) )
21		1le1 8491	. 2 |- 1 <_ 1
22		faccl 10669	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantr 274	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. NN )
24	23	nnred 8891	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. RR )
25		nn0re 9144	. . . . . . . 8 |- ( k e. NN0 -> k e. RR )
26	25	adantr 274	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k e. RR )
27		simpl 108	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k e. NN0 )
28	26, 27	reexpcld 10626	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) e. RR )
29		nn0p1nn 9174	. . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN )
30	29	adantr 274	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. NN )
31	30	nnred 8891	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. RR )
32	31, 27	reexpcld 10626	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( k + 1 ) ^ k ) e. RR )
33		simpr 109	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) <_ ( k ^ k ) )
34		nn0ge0 9160	. . . . . . . 8 |- ( k e. NN0 -> 0 <_ k )
35	34	adantr 274	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 <_ k )
36	26	lep1d 8847	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k <_ ( k + 1 ) )
37		leexp1a 10531	. . . . . . 7 |- ( ( ( k e. RR /\ ( k + 1 ) e. RR /\ k e. NN0 ) /\ ( 0 <_ k /\ k <_ ( k + 1 ) ) ) -> ( k ^ k ) <_ ( ( k + 1 ) ^ k ) )
38	26, 31, 27, 35, 36, 37	syl32anc 1241	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) <_ ( ( k + 1 ) ^ k ) )
39	24, 28, 32, 33, 38	letrd 8043	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) <_ ( ( k + 1 ) ^ k ) )
40	30	nngt0d 8922	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 < ( k + 1 ) )
41		lemul1 8512	. . . . . 6 |- ( ( ( ! ` k ) e. RR /\ ( ( k + 1 ) ^ k ) e. RR /\ ( ( k + 1 ) e. RR /\ 0 < ( k + 1 ) ) ) -> ( ( ! ` k ) <_ ( ( k + 1 ) ^ k ) <-> ( ( ! ` k ) x. ( k + 1 ) ) <_ ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) ) )
42	24, 32, 31, 40, 41	syl112anc 1237	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( ! ` k ) <_ ( ( k + 1 ) ^ k ) <-> ( ( ! ` k ) x. ( k + 1 ) ) <_ ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) ) )
43	39, 42	mpbid 146	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( ! ` k ) x. ( k + 1 ) ) <_ ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) )
44		facp1 10664	. . . . 5 |- ( k e. NN0 -> ( ! ` ( k + 1 ) ) = ( ( ! ` k ) x. ( k + 1 ) ) )
45	44	adantr 274	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` ( k + 1 ) ) = ( ( ! ` k ) x. ( k + 1 ) ) )
46	30	nncnd 8892	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. CC )
47	46, 27	expp1d 10610	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( k + 1 ) ^ ( k + 1 ) ) = ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) )
48	43, 45, 47	3brtr4d 4021	. . 3 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) )
49	48	ex 114	. 2 |- ( k e. NN0 -> ( ( ! ` k ) <_ ( k ^ k ) -> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) ) )
50	8, 12, 16, 20, 21, 49	nn0ind 9326	1 |- ( N e. NN0 -> ( ! ` N ) <_ ( N ^ N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   < clt 7954   <_ cle 7955   NNcn 8878   NN0cn0 9135   ^cexp 10475   !cfa 10659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476  df-fac 10660

This theorem is referenced by: (None)