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Theorem 2expltfac 13355
Description: The factorial grows faster than two to the power  N. (Contributed by Mario Carneiro, 15-Sep-2016.)
Assertion
Ref Expression
2expltfac  |-  ( N  e.  ( ZZ>= `  4
)  ->  ( 2 ^ N )  < 
( ! `  N
) )

Proof of Theorem 2expltfac
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6030 . . . 4  |-  ( x  =  4  ->  (
2 ^ x )  =  ( 2 ^ 4 ) )
2 2exp4 13350 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
31, 2syl6eq 2437 . . 3  |-  ( x  =  4  ->  (
2 ^ x )  = ; 1 6 )
4 fveq2 5670 . . . 4  |-  ( x  =  4  ->  ( ! `  x )  =  ( ! ` 
4 ) )
5 fac4 11503 . . . 4  |-  ( ! `
 4 )  = ; 2
4
64, 5syl6eq 2437 . . 3  |-  ( x  =  4  ->  ( ! `  x )  = ; 2 4 )
73, 6breq12d 4168 . 2  |-  ( x  =  4  ->  (
( 2 ^ x
)  <  ( ! `  x )  <-> ; 1 6  < ; 2 4 ) )
8 oveq2 6030 . . 3  |-  ( x  =  n  ->  (
2 ^ x )  =  ( 2 ^ n ) )
9 fveq2 5670 . . 3  |-  ( x  =  n  ->  ( ! `  x )  =  ( ! `  n ) )
108, 9breq12d 4168 . 2  |-  ( x  =  n  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ n )  < 
( ! `  n
) ) )
11 oveq2 6030 . . 3  |-  ( x  =  ( n  + 
1 )  ->  (
2 ^ x )  =  ( 2 ^ ( n  +  1 ) ) )
12 fveq2 5670 . . 3  |-  ( x  =  ( n  + 
1 )  ->  ( ! `  x )  =  ( ! `  ( n  +  1
) ) )
1311, 12breq12d 4168 . 2  |-  ( x  =  ( n  + 
1 )  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ ( n  + 
1 ) )  < 
( ! `  (
n  +  1 ) ) ) )
14 oveq2 6030 . . 3  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
15 fveq2 5670 . . 3  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1614, 15breq12d 4168 . 2  |-  ( x  =  N  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ N )  < 
( ! `  N
) ) )
17 1nn0 10171 . . . 4  |-  1  e.  NN0
18 2nn0 10172 . . . 4  |-  2  e.  NN0
19 6nn0 10176 . . . 4  |-  6  e.  NN0
20 4nn0 10174 . . . 4  |-  4  e.  NN0
21 6lt10 10115 . . . 4  |-  6  <  10
22 1lt2 10076 . . . 4  |-  1  <  2
2317, 18, 19, 20, 21, 22decltc 10338 . . 3  |- ; 1 6  < ; 2 4
2423a1i 11 . 2  |-  ( 4  e.  ZZ  -> ; 1 6  < ; 2 4 )
25 2nn 10067 . . . . . . . . 9  |-  2  e.  NN
2625a1i 11 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  NN )
27 4nn 10069 . . . . . . . . . 10  |-  4  e.  NN
28 simpl 444 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  ( ZZ>= ` 
4 ) )
29 nnuz 10455 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3029uztrn2 10437 . . . . . . . . . 10  |-  ( ( 4  e.  NN  /\  n  e.  ( ZZ>= ` 
4 ) )  ->  n  e.  NN )
3127, 28, 30sylancr 645 . . . . . . . . 9  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN )
3231nnnn0d 10208 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN0 )
3326, 32nnexpcld 11473 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  NN )
3433nnred 9949 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  RR )
35 2re 10003 . . . . . . 7  |-  2  e.  RR
3635a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR )
3734, 36remulcld 9051 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  e.  RR )
38 faccl 11505 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
3932, 38syl 16 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN )
4039nnred 9949 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  RR )
4140, 36remulcld 9051 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  e.  RR )
4231nnred 9949 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  RR )
43 1re 9025 . . . . . . . 8  |-  1  e.  RR
4443a1i 11 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  e.  RR )
4542, 44readdcld 9050 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( n  +  1 )  e.  RR )
4640, 45remulcld 9051 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  (
n  +  1 ) )  e.  RR )
47 2rp 10551 . . . . . . 7  |-  2  e.  RR+
4847a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR+ )
49 simpr 448 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  <  ( ! `  n ) )
5034, 40, 48, 49ltmul1dd 10633 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  2 ) )
5139nnnn0d 10208 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN0 )
5251nn0ge0d 10211 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
0  <_  ( ! `  n ) )
53 df-2 9992 . . . . . . 7  |-  2  =  ( 1  +  1 )
5431nnge1d 9976 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  <_  n )
5544, 42, 44, 54leadd1dd 9574 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 1  +  1 )  <_  ( n  +  1 ) )
5653, 55syl5eqbr 4188 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  <_  ( n  +  1 ) )
5736, 45, 40, 52, 56lemul2ad 9885 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  <_  ( ( ! `  n )  x.  ( n  +  1 ) ) )
5837, 41, 46, 50, 57ltletrd 9164 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
59 2cn 10004 . . . . . 6  |-  2  e.  CC
6059a1i 11 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  CC )
6160, 32expp1d 11453 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  =  ( ( 2 ^ n )  x.  2 ) )
62 facp1 11500 . . . . 5  |-  ( n  e.  NN0  ->  ( ! `
 ( n  + 
1 ) )  =  ( ( ! `  n )  x.  (
n  +  1 ) ) )
6332, 62syl 16 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  (
n  +  1 ) )  =  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
6458, 61, 633brtr4d 4185 . . 3  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  <  ( ! `
 ( n  + 
1 ) ) )
6564ex 424 . 2  |-  ( n  e.  ( ZZ>= `  4
)  ->  ( (
2 ^ n )  <  ( ! `  n )  ->  (
2 ^ ( n  +  1 ) )  <  ( ! `  ( n  +  1
) ) ) )
667, 10, 13, 16, 24, 65uzind4 10468 1  |-  ( N  e.  ( ZZ>= `  4
)  ->  ( 2 ^ N )  < 
( ! `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   1c1 8926    + caddc 8928    x. cmul 8930    < clt 9055    <_ cle 9056   NNcn 9934   2c2 9983   4c4 9985   6c6 9987   NN0cn0 10155   ZZcz 10216  ;cdc 10316   ZZ>=cuz 10422   RR+crp 10546   ^cexp 11311   !cfa 11495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312  df-fac 11496
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