ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  facubnd Unicode version

Theorem facubnd 9769
Description: An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
Assertion
Ref Expression
facubnd  |-  ( N  e.  NN0  ->  ( ! `
 N )  <_ 
( N ^ N
) )

Proof of Theorem facubnd
Dummy variables  m  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5209 . . . 4  |-  ( m  =  0  ->  ( ! `  m )  =  ( ! ` 
0 ) )
2 fac0 9752 . . . 4  |-  ( ! `
 0 )  =  1
31, 2syl6eq 2130 . . 3  |-  ( m  =  0  ->  ( ! `  m )  =  1 )
4 id 19 . . . . 5  |-  ( m  =  0  ->  m  =  0 )
54, 4oveq12d 5561 . . . 4  |-  ( m  =  0  ->  (
m ^ m )  =  ( 0 ^ 0 ) )
6 0exp0e1 9578 . . . 4  |-  ( 0 ^ 0 )  =  1
75, 6syl6eq 2130 . . 3  |-  ( m  =  0  ->  (
m ^ m )  =  1 )
83, 7breq12d 3806 . 2  |-  ( m  =  0  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  1  <_  1 ) )
9 fveq2 5209 . . 3  |-  ( m  =  k  ->  ( ! `  m )  =  ( ! `  k ) )
10 id 19 . . . 4  |-  ( m  =  k  ->  m  =  k )
1110, 10oveq12d 5561 . . 3  |-  ( m  =  k  ->  (
m ^ m )  =  ( k ^
k ) )
129, 11breq12d 3806 . 2  |-  ( m  =  k  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  k )  <_  (
k ^ k ) ) )
13 fveq2 5209 . . 3  |-  ( m  =  ( k  +  1 )  ->  ( ! `  m )  =  ( ! `  ( k  +  1 ) ) )
14 id 19 . . . 4  |-  ( m  =  ( k  +  1 )  ->  m  =  ( k  +  1 ) )
1514, 14oveq12d 5561 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
m ^ m )  =  ( ( k  +  1 ) ^
( k  +  1 ) ) )
1613, 15breq12d 3806 . 2  |-  ( m  =  ( k  +  1 )  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  ( k  +  1 ) )  <_  (
( k  +  1 ) ^ ( k  +  1 ) ) ) )
17 fveq2 5209 . . 3  |-  ( m  =  N  ->  ( ! `  m )  =  ( ! `  N ) )
18 id 19 . . . 4  |-  ( m  =  N  ->  m  =  N )
1918, 18oveq12d 5561 . . 3  |-  ( m  =  N  ->  (
m ^ m )  =  ( N ^ N ) )
2017, 19breq12d 3806 . 2  |-  ( m  =  N  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  N )  <_  ( N ^ N ) ) )
21 1le1 7739 . 2  |-  1  <_  1
22 faccl 9759 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2322adantr 270 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  e.  NN )
2423nnred 8119 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  e.  RR )
25 nn0re 8364 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  RR )
2625adantr 270 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  e.  RR )
27 simpl 107 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  e.  NN0 )
2826, 27reexpcld 9719 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k ^ k
)  e.  RR )
29 nn0p1nn 8394 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3029adantr 270 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  NN )
3130nnred 8119 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  RR )
3231, 27reexpcld 9719 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( k  +  1 ) ^ k
)  e.  RR )
33 simpr 108 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  <_  ( k ^ k ) )
34 nn0ge0 8380 . . . . . . . 8  |-  ( k  e.  NN0  ->  0  <_ 
k )
3534adantr 270 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
0  <_  k )
3626lep1d 8076 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  <_  ( k  +  1 ) )
37 leexp1a 9628 . . . . . . 7  |-  ( ( ( k  e.  RR  /\  ( k  +  1 )  e.  RR  /\  k  e.  NN0 )  /\  ( 0  <_  k  /\  k  <_  ( k  +  1 ) ) )  ->  ( k ^ k )  <_ 
( ( k  +  1 ) ^ k
) )
3826, 31, 27, 35, 36, 37syl32anc 1178 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k ^ k
)  <_  ( (
k  +  1 ) ^ k ) )
3924, 28, 32, 33, 38letrd 7300 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  <_  ( (
k  +  1 ) ^ k ) )
4030nngt0d 8149 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
0  <  ( k  +  1 ) )
41 lemul1 7760 . . . . . 6  |-  ( ( ( ! `  k
)  e.  RR  /\  ( ( k  +  1 ) ^ k
)  e.  RR  /\  ( ( k  +  1 )  e.  RR  /\  0  <  ( k  +  1 ) ) )  ->  ( ( ! `  k )  <_  ( ( k  +  1 ) ^ k
)  <->  ( ( ! `
 k )  x.  ( k  +  1 ) )  <_  (
( ( k  +  1 ) ^ k
)  x.  ( k  +  1 ) ) ) )
4224, 32, 31, 40, 41syl112anc 1174 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( ! `  k )  <_  (
( k  +  1 ) ^ k )  <-> 
( ( ! `  k )  x.  (
k  +  1 ) )  <_  ( (
( k  +  1 ) ^ k )  x.  ( k  +  1 ) ) ) )
4339, 42mpbid 145 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( ! `  k )  x.  (
k  +  1 ) )  <_  ( (
( k  +  1 ) ^ k )  x.  ( k  +  1 ) ) )
44 facp1 9754 . . . . 5  |-  ( k  e.  NN0  ->  ( ! `
 ( k  +  1 ) )  =  ( ( ! `  k )  x.  (
k  +  1 ) ) )
4544adantr 270 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  (
k  +  1 ) )  =  ( ( ! `  k )  x.  ( k  +  1 ) ) )
4630nncnd 8120 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  CC )
4746, 27expp1d 9703 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( k  +  1 ) ^ (
k  +  1 ) )  =  ( ( ( k  +  1 ) ^ k )  x.  ( k  +  1 ) ) )
4843, 45, 473brtr4d 3823 . . 3  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  (
k  +  1 ) )  <_  ( (
k  +  1 ) ^ ( k  +  1 ) ) )
4948ex 113 . 2  |-  ( k  e.  NN0  ->  ( ( ! `  k )  <_  ( k ^
k )  ->  ( ! `  ( k  +  1 ) )  <_  ( ( k  +  1 ) ^
( k  +  1 ) ) ) )
508, 12, 16, 20, 21, 49nn0ind 8542 1  |-  ( N  e.  NN0  ->  ( ! `
 N )  <_ 
( N ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   RRcr 7042   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048    < clt 7215    <_ cle 7216   NNcn 8106   NN0cn0 8355   ^cexp 9572   !cfa 9749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522  df-iexp 9573  df-fac 9750
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator