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Theorem facubnd 12358
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 5872 . . . 4  |-  ( m  =  0  ->  ( ! `  m )  =  ( ! ` 
0 ) )
2 fac0 12336 . . . 4  |-  ( ! `
 0 )  =  1
31, 2syl6eq 2524 . . 3  |-  ( m  =  0  ->  ( ! `  m )  =  1 )
4 id 22 . . . . 5  |-  ( m  =  0  ->  m  =  0 )
54, 4oveq12d 6313 . . . 4  |-  ( m  =  0  ->  (
m ^ m )  =  ( 0 ^ 0 ) )
6 0exp0e1 12151 . . . 4  |-  ( 0 ^ 0 )  =  1
75, 6syl6eq 2524 . . 3  |-  ( m  =  0  ->  (
m ^ m )  =  1 )
83, 7breq12d 4466 . 2  |-  ( m  =  0  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  1  <_  1 ) )
9 fveq2 5872 . . 3  |-  ( m  =  k  ->  ( ! `  m )  =  ( ! `  k ) )
10 id 22 . . . 4  |-  ( m  =  k  ->  m  =  k )
1110, 10oveq12d 6313 . . 3  |-  ( m  =  k  ->  (
m ^ m )  =  ( k ^
k ) )
129, 11breq12d 4466 . 2  |-  ( m  =  k  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  k )  <_  (
k ^ k ) ) )
13 fveq2 5872 . . 3  |-  ( m  =  ( k  +  1 )  ->  ( ! `  m )  =  ( ! `  ( k  +  1 ) ) )
14 id 22 . . . 4  |-  ( m  =  ( k  +  1 )  ->  m  =  ( k  +  1 ) )
1514, 14oveq12d 6313 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
m ^ m )  =  ( ( k  +  1 ) ^
( k  +  1 ) ) )
1613, 15breq12d 4466 . 2  |-  ( m  =  ( k  +  1 )  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  ( k  +  1 ) )  <_  (
( k  +  1 ) ^ ( k  +  1 ) ) ) )
17 fveq2 5872 . . 3  |-  ( m  =  N  ->  ( ! `  m )  =  ( ! `  N ) )
18 id 22 . . . 4  |-  ( m  =  N  ->  m  =  N )
1918, 18oveq12d 6313 . . 3  |-  ( m  =  N  ->  (
m ^ m )  =  ( N ^ N ) )
2017, 19breq12d 4466 . 2  |-  ( m  =  N  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  N )  <_  ( N ^ N ) ) )
21 1le1 10189 . 2  |-  1  <_  1
22 faccl 12343 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2322adantr 465 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  e.  NN )
2423nnred 10563 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  e.  RR )
25 nn0re 10816 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  RR )
2625adantr 465 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  e.  RR )
27 simpl 457 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  e.  NN0 )
2826, 27reexpcld 12307 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k ^ k
)  e.  RR )
29 nn0p1nn 10847 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3029adantr 465 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  NN )
3130nnred 10563 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  RR )
3231, 27reexpcld 12307 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( k  +  1 ) ^ k
)  e.  RR )
33 simpr 461 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  <_  ( k ^ k ) )
34 nn0ge0 10833 . . . . . . . 8  |-  ( k  e.  NN0  ->  0  <_ 
k )
3534adantr 465 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
0  <_  k )
3626lep1d 10489 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  <_  ( k  +  1 ) )
37 leexp1a 12204 . . . . . . 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 1236 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k ^ k
)  <_  ( (
k  +  1 ) ^ k ) )
3924, 28, 32, 33, 38letrd 9750 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  <_  ( (
k  +  1 ) ^ k ) )
4030nngt0d 10591 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
0  <  ( k  +  1 ) )
41 lemul1 10406 . . . . . 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 1232 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( ! `  k )  <_  (
( k  +  1 ) ^ k )  <-> 
( ( ! `  k )  x.  (
k  +  1 ) )  <_  ( (
( k  +  1 ) ^ k )  x.  ( k  +  1 ) ) ) )
4339, 42mpbid 210 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( ! `  k )  x.  (
k  +  1 ) )  <_  ( (
( k  +  1 ) ^ k )  x.  ( k  +  1 ) ) )
44 facp1 12338 . . . . 5  |-  ( k  e.  NN0  ->  ( ! `
 ( k  +  1 ) )  =  ( ( ! `  k )  x.  (
k  +  1 ) ) )
4544adantr 465 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  (
k  +  1 ) )  =  ( ( ! `  k )  x.  ( k  +  1 ) ) )
4630nncnd 10564 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  CC )
4746, 27expp1d 12291 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( k  +  1 ) ^ (
k  +  1 ) )  =  ( ( ( k  +  1 ) ^ k )  x.  ( k  +  1 ) ) )
4843, 45, 473brtr4d 4483 . . 3  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  (
k  +  1 ) )  <_  ( (
k  +  1 ) ^ ( k  +  1 ) ) )
4948ex 434 . 2  |-  ( k  e.  NN0  ->  ( ( ! `  k )  <_  ( k ^
k )  ->  ( ! `  ( k  +  1 ) )  <_  ( ( k  +  1 ) ^
( k  +  1 ) ) ) )
508, 12, 16, 20, 21, 49nn0ind 10969 1  |-  ( N  e.  NN0  ->  ( ! `
 N )  <_ 
( N ^ N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    <_ cle 9641   NNcn 10548   NN0cn0 10807   ^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-n0 10808  df-z 10877  df-uz 11095  df-seq 12088  df-exp 12147  df-fac 12334
This theorem is referenced by:  logfacubnd  23362  pgrple2abl  32438
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