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Theorem 2expltfac 15048
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 6309 . . . 4  |-  ( x  =  4  ->  (
2 ^ x )  =  ( 2 ^ 4 ) )
2 2exp4 15042 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
31, 2syl6eq 2479 . . 3  |-  ( x  =  4  ->  (
2 ^ x )  = ; 1 6 )
4 fveq2 5877 . . . 4  |-  ( x  =  4  ->  ( ! `  x )  =  ( ! ` 
4 ) )
5 fac4 12466 . . . 4  |-  ( ! `
 4 )  = ; 2
4
64, 5syl6eq 2479 . . 3  |-  ( x  =  4  ->  ( ! `  x )  = ; 2 4 )
73, 6breq12d 4433 . 2  |-  ( x  =  4  ->  (
( 2 ^ x
)  <  ( ! `  x )  <-> ; 1 6  < ; 2 4 ) )
8 oveq2 6309 . . 3  |-  ( x  =  n  ->  (
2 ^ x )  =  ( 2 ^ n ) )
9 fveq2 5877 . . 3  |-  ( x  =  n  ->  ( ! `  x )  =  ( ! `  n ) )
108, 9breq12d 4433 . 2  |-  ( x  =  n  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ n )  < 
( ! `  n
) ) )
11 oveq2 6309 . . 3  |-  ( x  =  ( n  + 
1 )  ->  (
2 ^ x )  =  ( 2 ^ ( n  +  1 ) ) )
12 fveq2 5877 . . 3  |-  ( x  =  ( n  + 
1 )  ->  ( ! `  x )  =  ( ! `  ( n  +  1
) ) )
1311, 12breq12d 4433 . 2  |-  ( x  =  ( n  + 
1 )  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ ( n  + 
1 ) )  < 
( ! `  (
n  +  1 ) ) ) )
14 oveq2 6309 . . 3  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
15 fveq2 5877 . . 3  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1614, 15breq12d 4433 . 2  |-  ( x  =  N  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ N )  < 
( ! `  N
) ) )
17 1nn0 10885 . . . 4  |-  1  e.  NN0
18 2nn0 10886 . . . 4  |-  2  e.  NN0
19 6nn0 10890 . . . 4  |-  6  e.  NN0
20 4nn0 10888 . . . 4  |-  4  e.  NN0
21 6lt10 10815 . . . 4  |-  6  <  10
22 1lt2 10776 . . . 4  |-  1  <  2
2317, 18, 19, 20, 21, 22decltc 11073 . . 3  |- ; 1 6  < ; 2 4
2423a1i 11 . 2  |-  ( 4  e.  ZZ  -> ; 1 6  < ; 2 4 )
25 2nn 10767 . . . . . . . . 9  |-  2  e.  NN
2625a1i 11 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  NN )
27 4nn 10769 . . . . . . . . . 10  |-  4  e.  NN
28 simpl 458 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  ( ZZ>= ` 
4 ) )
29 eluznn 11229 . . . . . . . . . 10  |-  ( ( 4  e.  NN  /\  n  e.  ( ZZ>= ` 
4 ) )  ->  n  e.  NN )
3027, 28, 29sylancr 667 . . . . . . . . 9  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN )
3130nnnn0d 10925 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN0 )
3226, 31nnexpcld 12436 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  NN )
3332nnred 10624 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  RR )
34 2re 10679 . . . . . . 7  |-  2  e.  RR
3534a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR )
3633, 35remulcld 9671 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  e.  RR )
37 faccl 12468 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
3831, 37syl 17 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN )
3938nnred 10624 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  RR )
4039, 35remulcld 9671 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  e.  RR )
4130nnred 10624 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  RR )
42 1red 9658 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  e.  RR )
4341, 42readdcld 9670 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( n  +  1 )  e.  RR )
4439, 43remulcld 9671 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  (
n  +  1 ) )  e.  RR )
45 2rp 11307 . . . . . . 7  |-  2  e.  RR+
4645a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR+ )
47 simpr 462 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  <  ( ! `  n ) )
4833, 39, 46, 47ltmul1dd 11393 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  2 ) )
4938nnnn0d 10925 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN0 )
5049nn0ge0d 10928 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
0  <_  ( ! `  n ) )
51 df-2 10668 . . . . . . 7  |-  2  =  ( 1  +  1 )
5230nnge1d 10652 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  <_  n )
5342, 41, 42, 52leadd1dd 10227 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 1  +  1 )  <_  ( n  +  1 ) )
5451, 53syl5eqbr 4454 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  <_  ( n  +  1 ) )
5535, 43, 39, 50, 54lemul2ad 10547 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  <_  ( ( ! `  n )  x.  ( n  +  1 ) ) )
5636, 40, 44, 48, 55ltletrd 9795 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
57 2cnd 10682 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  CC )
5857, 31expp1d 12416 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  =  ( ( 2 ^ n )  x.  2 ) )
59 facp1 12463 . . . . 5  |-  ( n  e.  NN0  ->  ( ! `
 ( n  + 
1 ) )  =  ( ( ! `  n )  x.  (
n  +  1 ) ) )
6031, 59syl 17 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  (
n  +  1 ) )  =  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
6156, 58, 603brtr4d 4451 . . 3  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  <  ( ! `
 ( n  + 
1 ) ) )
6261ex 435 . 2  |-  ( n  e.  ( ZZ>= `  4
)  ->  ( (
2 ^ n )  <  ( ! `  n )  ->  (
2 ^ ( n  +  1 ) )  <  ( ! `  ( n  +  1
) ) ) )
637, 10, 13, 16, 24, 62uzind4 11217 1  |-  ( N  e.  ( ZZ>= `  4
)  ->  ( 2 ^ N )  < 
( ! `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   RRcr 9538   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676   NNcn 10609   2c2 10659   4c4 10661   6c6 10663   NN0cn0 10869   ZZcz 10937  ;cdc 11051   ZZ>=cuz 11159   RR+crp 11302   ^cexp 12271   !cfa 12458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-seq 12213  df-exp 12272  df-fac 12459
This theorem is referenced by: (None)
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