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Theorem 2expltfac 14452
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 6303 . . . 4  |-  ( x  =  4  ->  (
2 ^ x )  =  ( 2 ^ 4 ) )
2 2exp4 14447 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
31, 2syl6eq 2524 . . 3  |-  ( x  =  4  ->  (
2 ^ x )  = ; 1 6 )
4 fveq2 5872 . . . 4  |-  ( x  =  4  ->  ( ! `  x )  =  ( ! ` 
4 ) )
5 fac4 12341 . . . 4  |-  ( ! `
 4 )  = ; 2
4
64, 5syl6eq 2524 . . 3  |-  ( x  =  4  ->  ( ! `  x )  = ; 2 4 )
73, 6breq12d 4466 . 2  |-  ( x  =  4  ->  (
( 2 ^ x
)  <  ( ! `  x )  <-> ; 1 6  < ; 2 4 ) )
8 oveq2 6303 . . 3  |-  ( x  =  n  ->  (
2 ^ x )  =  ( 2 ^ n ) )
9 fveq2 5872 . . 3  |-  ( x  =  n  ->  ( ! `  x )  =  ( ! `  n ) )
108, 9breq12d 4466 . 2  |-  ( x  =  n  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ n )  < 
( ! `  n
) ) )
11 oveq2 6303 . . 3  |-  ( x  =  ( n  + 
1 )  ->  (
2 ^ x )  =  ( 2 ^ ( n  +  1 ) ) )
12 fveq2 5872 . . 3  |-  ( x  =  ( n  + 
1 )  ->  ( ! `  x )  =  ( ! `  ( n  +  1
) ) )
1311, 12breq12d 4466 . 2  |-  ( x  =  ( n  + 
1 )  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ ( n  + 
1 ) )  < 
( ! `  (
n  +  1 ) ) ) )
14 oveq2 6303 . . 3  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
15 fveq2 5872 . . 3  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1614, 15breq12d 4466 . 2  |-  ( x  =  N  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ N )  < 
( ! `  N
) ) )
17 1nn0 10823 . . . 4  |-  1  e.  NN0
18 2nn0 10824 . . . 4  |-  2  e.  NN0
19 6nn0 10828 . . . 4  |-  6  e.  NN0
20 4nn0 10826 . . . 4  |-  4  e.  NN0
21 6lt10 10753 . . . 4  |-  6  <  10
22 1lt2 10714 . . . 4  |-  1  <  2
2317, 18, 19, 20, 21, 22decltc 11010 . . 3  |- ; 1 6  < ; 2 4
2423a1i 11 . 2  |-  ( 4  e.  ZZ  -> ; 1 6  < ; 2 4 )
25 2nn 10705 . . . . . . . . 9  |-  2  e.  NN
2625a1i 11 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  NN )
27 4nn 10707 . . . . . . . . . 10  |-  4  e.  NN
28 simpl 457 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  ( ZZ>= ` 
4 ) )
29 eluznn 11164 . . . . . . . . . 10  |-  ( ( 4  e.  NN  /\  n  e.  ( ZZ>= ` 
4 ) )  ->  n  e.  NN )
3027, 28, 29sylancr 663 . . . . . . . . 9  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN )
3130nnnn0d 10864 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN0 )
3226, 31nnexpcld 12311 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  NN )
3332nnred 10563 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  RR )
34 2re 10617 . . . . . . 7  |-  2  e.  RR
3534a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR )
3633, 35remulcld 9636 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  e.  RR )
37 faccl 12343 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
3831, 37syl 16 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN )
3938nnred 10563 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  RR )
4039, 35remulcld 9636 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  e.  RR )
4130nnred 10563 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  RR )
42 1red 9623 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  e.  RR )
4341, 42readdcld 9635 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( n  +  1 )  e.  RR )
4439, 43remulcld 9636 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  (
n  +  1 ) )  e.  RR )
45 2rp 11237 . . . . . . 7  |-  2  e.  RR+
4645a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR+ )
47 simpr 461 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  <  ( ! `  n ) )
4833, 39, 46, 47ltmul1dd 11319 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  2 ) )
4938nnnn0d 10864 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN0 )
5049nn0ge0d 10867 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
0  <_  ( ! `  n ) )
51 df-2 10606 . . . . . . 7  |-  2  =  ( 1  +  1 )
5230nnge1d 10590 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  <_  n )
5342, 41, 42, 52leadd1dd 10178 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 1  +  1 )  <_  ( n  +  1 ) )
5451, 53syl5eqbr 4486 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  <_  ( n  +  1 ) )
5535, 43, 39, 50, 54lemul2ad 10498 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  <_  ( ( ! `  n )  x.  ( n  +  1 ) ) )
5636, 40, 44, 48, 55ltletrd 9753 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
57 2cnd 10620 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  CC )
5857, 31expp1d 12291 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  =  ( ( 2 ^ n )  x.  2 ) )
59 facp1 12338 . . . . 5  |-  ( n  e.  NN0  ->  ( ! `
 ( n  + 
1 ) )  =  ( ( ! `  n )  x.  (
n  +  1 ) ) )
6031, 59syl 16 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  (
n  +  1 ) )  =  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
6156, 58, 603brtr4d 4483 . . 3  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  <  ( ! `
 ( n  + 
1 ) ) )
6261ex 434 . 2  |-  ( n  e.  ( ZZ>= `  4
)  ->  ( (
2 ^ n )  <  ( ! `  n )  ->  (
2 ^ ( n  +  1 ) )  <  ( ! `  ( n  +  1
) ) ) )
637, 10, 13, 16, 24, 62uzind4 11151 1  |-  ( N  e.  ( ZZ>= `  4
)  ->  ( 2 ^ N )  < 
( ! `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   RRcr 9503   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    <_ cle 9641   NNcn 10548   2c2 10597   4c4 10599   6c6 10601   NN0cn0 10807   ZZcz 10876  ;cdc 10988   ZZ>=cuz 11094   RR+crp 11232   ^cexp 12146   !cfa 12333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-seq 12088  df-exp 12147  df-fac 12334
This theorem is referenced by: (None)
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