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Theorem 2expltfac 14241
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 6211 . . . 4  |-  ( x  =  4  ->  (
2 ^ x )  =  ( 2 ^ 4 ) )
2 2exp4 14236 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
31, 2syl6eq 2511 . . 3  |-  ( x  =  4  ->  (
2 ^ x )  = ; 1 6 )
4 fveq2 5802 . . . 4  |-  ( x  =  4  ->  ( ! `  x )  =  ( ! ` 
4 ) )
5 fac4 12180 . . . 4  |-  ( ! `
 4 )  = ; 2
4
64, 5syl6eq 2511 . . 3  |-  ( x  =  4  ->  ( ! `  x )  = ; 2 4 )
73, 6breq12d 4416 . 2  |-  ( x  =  4  ->  (
( 2 ^ x
)  <  ( ! `  x )  <-> ; 1 6  < ; 2 4 ) )
8 oveq2 6211 . . 3  |-  ( x  =  n  ->  (
2 ^ x )  =  ( 2 ^ n ) )
9 fveq2 5802 . . 3  |-  ( x  =  n  ->  ( ! `  x )  =  ( ! `  n ) )
108, 9breq12d 4416 . 2  |-  ( x  =  n  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ n )  < 
( ! `  n
) ) )
11 oveq2 6211 . . 3  |-  ( x  =  ( n  + 
1 )  ->  (
2 ^ x )  =  ( 2 ^ ( n  +  1 ) ) )
12 fveq2 5802 . . 3  |-  ( x  =  ( n  + 
1 )  ->  ( ! `  x )  =  ( ! `  ( n  +  1
) ) )
1311, 12breq12d 4416 . 2  |-  ( x  =  ( n  + 
1 )  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ ( n  + 
1 ) )  < 
( ! `  (
n  +  1 ) ) ) )
14 oveq2 6211 . . 3  |-  ( x  =  N  ->  (
2 ^ x )  =  ( 2 ^ N ) )
15 fveq2 5802 . . 3  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1614, 15breq12d 4416 . 2  |-  ( x  =  N  ->  (
( 2 ^ x
)  <  ( ! `  x )  <->  ( 2 ^ N )  < 
( ! `  N
) ) )
17 1nn0 10710 . . . 4  |-  1  e.  NN0
18 2nn0 10711 . . . 4  |-  2  e.  NN0
19 6nn0 10715 . . . 4  |-  6  e.  NN0
20 4nn0 10713 . . . 4  |-  4  e.  NN0
21 6lt10 10642 . . . 4  |-  6  <  10
22 1lt2 10603 . . . 4  |-  1  <  2
2317, 18, 19, 20, 21, 22decltc 10892 . . 3  |- ; 1 6  < ; 2 4
2423a1i 11 . 2  |-  ( 4  e.  ZZ  -> ; 1 6  < ; 2 4 )
25 2nn 10594 . . . . . . . . 9  |-  2  e.  NN
2625a1i 11 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  NN )
27 4nn 10596 . . . . . . . . . 10  |-  4  e.  NN
28 simpl 457 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  ( ZZ>= ` 
4 ) )
29 eluznn 11040 . . . . . . . . . 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 10751 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  NN0 )
3226, 31nnexpcld 12150 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  NN )
3332nnred 10452 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ n
)  e.  RR )
34 2re 10506 . . . . . . 7  |-  2  e.  RR
3534a1i 11 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  RR )
3633, 35remulcld 9529 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  e.  RR )
37 faccl 12182 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ! `
 n )  e.  NN )
3831, 37syl 16 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN )
3938nnred 10452 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  RR )
4039, 35remulcld 9529 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  e.  RR )
4130nnred 10452 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  ->  n  e.  RR )
42 1red 9516 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  e.  RR )
4341, 42readdcld 9528 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( n  +  1 )  e.  RR )
4439, 43remulcld 9529 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  (
n  +  1 ) )  e.  RR )
45 2rp 11111 . . . . . . 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 11193 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  2 ) )
4938nnnn0d 10751 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ! `  n
)  e.  NN0 )
5049nn0ge0d 10754 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
0  <_  ( ! `  n ) )
51 df-2 10495 . . . . . . 7  |-  2  =  ( 1  +  1 )
5230nnge1d 10479 . . . . . . . 8  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
1  <_  n )
5342, 41, 42, 52leadd1dd 10068 . . . . . . 7  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 1  +  1 )  <_  ( n  +  1 ) )
5451, 53syl5eqbr 4436 . . . . . 6  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  <_  ( n  +  1 ) )
5535, 43, 39, 50, 54lemul2ad 10388 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( ! `  n )  x.  2 )  <_  ( ( ! `  n )  x.  ( n  +  1 ) ) )
5636, 40, 44, 48, 55ltletrd 9646 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( ( 2 ^ n )  x.  2 )  <  ( ( ! `  n )  x.  ( n  + 
1 ) ) )
57 2cnd 10509 . . . . 5  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
2  e.  CC )
5857, 31expp1d 12130 . . . 4  |-  ( ( n  e.  ( ZZ>= ` 
4 )  /\  (
2 ^ n )  <  ( ! `  n ) )  -> 
( 2 ^ (
n  +  1 ) )  =  ( ( 2 ^ n )  x.  2 ) )
59 facp1 12177 . . . . 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 4433 . . 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 11027 1  |-  ( N  e.  ( ZZ>= `  4
)  ->  ( 2 ^ N )  < 
( ! `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   RRcr 9396   1c1 9398    + caddc 9400    x. cmul 9402    < clt 9533    <_ cle 9534   NNcn 10437   2c2 10486   4c4 10488   6c6 10490   NN0cn0 10694   ZZcz 10761  ;cdc 10870   ZZ>=cuz 10976   RR+crp 11106   ^cexp 11986   !cfa 12172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-rp 11107  df-seq 11928  df-exp 11987  df-fac 12173
This theorem is referenced by: (None)
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