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Theorem 2expltfac 13385
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 6052 . . . 4  |-  ( x  =  4  ->  (
2 ^ x )  =  ( 2 ^ 4 ) )
2 2exp4 13380 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
31, 2syl6eq 2456 . . 3  |-  ( x  =  4  ->  (
2 ^ x )  = ; 1 6 )
4 fveq2 5691 . . . 4  |-  ( x  =  4  ->  ( ! `  x )  =  ( ! ` 
4 ) )
5 fac4 11533 . . . 4  |-  ( ! `
 4 )  = ; 2
4
64, 5syl6eq 2456 . . 3  |-  ( x  =  4  ->  ( ! `  x )  = ; 2 4 )
73, 6breq12d 4189 . 2  |-  ( x  =  4  ->  (
( 2 ^ x
)  <  ( ! `  x )  <-> ; 1 6  < ; 2 4 ) )
8 oveq2 6052 . . 3  |-  ( x  =  n  ->  (
2 ^ x )  =  ( 2 ^ n ) )
9 fveq2 5691 . . 3  |-  ( x  =  n  ->  ( ! `  x )  =  ( ! `  n ) )
108, 9breq12d 4189 . 2  |-  ( x  =  n  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ n )  < 
( ! `  n
) ) )
11 oveq2 6052 . . 3  |-  ( x  =  ( n  + 
1 )  ->  (
2 ^ x )  =  ( 2 ^ ( n  +  1 ) ) )
12 fveq2 5691 . . 3  |-  ( x  =  ( n  + 
1 )  ->  ( ! `  x )  =  ( ! `  ( n  +  1
) ) )
1311, 12breq12d 4189 . 2  |-  ( x  =  ( n  + 
1 )  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ ( n  + 
1 ) )  < 
( ! `  (
n  +  1 ) ) ) )
14 oveq2 6052 . . 3  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
15 fveq2 5691 . . 3  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1614, 15breq12d 4189 . 2  |-  ( x  =  N  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ N )  < 
( ! `  N
) ) )
17 1nn0 10197 . . . 4  |-  1  e.  NN0
18 2nn0 10198 . . . 4  |-  2  e.  NN0
19 6nn0 10202 . . . 4  |-  6  e.  NN0
20 4nn0 10200 . . . 4  |-  4  e.  NN0
21 6lt10 10141 . . . 4  |-  6  <  10
22 1lt2 10102 . . . 4  |-  1  <  2
2317, 18, 19, 20, 21, 22decltc 10364 . . 3  |- ; 1 6  < ; 2 4
2423a1i 11 . 2  |-  ( 4  e.  ZZ  -> ; 1 6  < ; 2 4 )
25 2nn 10093 . . . . . . . . 9  |-  2  e.  NN
2625a1i 11 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  NN )
27 4nn 10095 . . . . . . . . . 10  |-  4  e.  NN
28 simpl 444 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  ( ZZ>= ` 
4 ) )
29 nnuz 10481 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3029uztrn2 10463 . . . . . . . . . 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 10234 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN0 )
3326, 32nnexpcld 11503 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  NN )
3433nnred 9975 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  RR )
35 2re 10029 . . . . . . 7  |-  2  e.  RR
3635a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR )
3734, 36remulcld 9076 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  e.  RR )
38 faccl 11535 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
3932, 38syl 16 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN )
4039nnred 9975 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  RR )
4140, 36remulcld 9076 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  e.  RR )
4231nnred 9975 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  RR )
43 1re 9050 . . . . . . . 8  |-  1  e.  RR
4443a1i 11 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  e.  RR )
4542, 44readdcld 9075 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( n  +  1 )  e.  RR )
4640, 45remulcld 9076 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  (
n  +  1 ) )  e.  RR )
47 2rp 10577 . . . . . . 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 10659 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  2 ) )
5139nnnn0d 10234 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN0 )
5251nn0ge0d 10237 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
0  <_  ( ! `  n ) )
53 df-2 10018 . . . . . . 7  |-  2  =  ( 1  +  1 )
5431nnge1d 10002 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  <_  n )
5544, 42, 44, 54leadd1dd 9600 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 1  +  1 )  <_  ( n  +  1 ) )
5653, 55syl5eqbr 4209 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  <_  ( n  +  1 ) )
5736, 45, 40, 52, 56lemul2ad 9911 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  <_  ( ( ! `  n )  x.  ( n  +  1 ) ) )
5837, 41, 46, 50, 57ltletrd 9190 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
59 2cn 10030 . . . . . 6  |-  2  e.  CC
6059a1i 11 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  CC )
6160, 32expp1d 11483 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  =  ( ( 2 ^ n )  x.  2 ) )
62 facp1 11530 . . . . 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 4206 . . 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 10494 1  |-  ( N  e.  ( ZZ>= `  4
)  ->  ( 2 ^ N )  < 
( ! `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   CCcc 8948   RRcr 8949   1c1 8951    + caddc 8953    x. cmul 8955    < clt 9080    <_ cle 9081   NNcn 9960   2c2 10009   4c4 10011   6c6 10013   NN0cn0 10181   ZZcz 10242  ;cdc 10342   ZZ>=cuz 10448   RR+crp 10572   ^cexp 11341   !cfa 11525
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-rp 10573  df-seq 11283  df-exp 11342  df-fac 11526
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