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Theorem facubnd 12280
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 5774 . . . 4  |-  ( m  =  0  ->  ( ! `  m )  =  ( ! ` 
0 ) )
2 fac0 12258 . . . 4  |-  ( ! `
 0 )  =  1
31, 2syl6eq 2439 . . 3  |-  ( m  =  0  ->  ( ! `  m )  =  1 )
4 id 22 . . . . 5  |-  ( m  =  0  ->  m  =  0 )
54, 4oveq12d 6214 . . . 4  |-  ( m  =  0  ->  (
m ^ m )  =  ( 0 ^ 0 ) )
6 0exp0e1 12074 . . . 4  |-  ( 0 ^ 0 )  =  1
75, 6syl6eq 2439 . . 3  |-  ( m  =  0  ->  (
m ^ m )  =  1 )
83, 7breq12d 4380 . 2  |-  ( m  =  0  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  1  <_  1 ) )
9 fveq2 5774 . . 3  |-  ( m  =  k  ->  ( ! `  m )  =  ( ! `  k ) )
10 id 22 . . . 4  |-  ( m  =  k  ->  m  =  k )
1110, 10oveq12d 6214 . . 3  |-  ( m  =  k  ->  (
m ^ m )  =  ( k ^
k ) )
129, 11breq12d 4380 . 2  |-  ( m  =  k  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  k )  <_  (
k ^ k ) ) )
13 fveq2 5774 . . 3  |-  ( m  =  ( k  +  1 )  ->  ( ! `  m )  =  ( ! `  ( k  +  1 ) ) )
14 id 22 . . . 4  |-  ( m  =  ( k  +  1 )  ->  m  =  ( k  +  1 ) )
1514, 14oveq12d 6214 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
m ^ m )  =  ( ( k  +  1 ) ^
( k  +  1 ) ) )
1613, 15breq12d 4380 . 2  |-  ( m  =  ( k  +  1 )  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  ( k  +  1 ) )  <_  (
( k  +  1 ) ^ ( k  +  1 ) ) ) )
17 fveq2 5774 . . 3  |-  ( m  =  N  ->  ( ! `  m )  =  ( ! `  N ) )
18 id 22 . . . 4  |-  ( m  =  N  ->  m  =  N )
1918, 18oveq12d 6214 . . 3  |-  ( m  =  N  ->  (
m ^ m )  =  ( N ^ N ) )
2017, 19breq12d 4380 . 2  |-  ( m  =  N  ->  (
( ! `  m
)  <_  ( m ^ m )  <->  ( ! `  N )  <_  ( N ^ N ) ) )
21 1le1 10094 . 2  |-  1  <_  1
22 faccl 12265 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2322adantr 463 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  e.  NN )
2423nnred 10467 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  e.  RR )
25 nn0re 10721 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  RR )
2625adantr 463 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  e.  RR )
27 simpl 455 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  e.  NN0 )
2826, 27reexpcld 12229 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k ^ k
)  e.  RR )
29 nn0p1nn 10752 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3029adantr 463 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  NN )
3130nnred 10467 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  RR )
3231, 27reexpcld 12229 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( k  +  1 ) ^ k
)  e.  RR )
33 simpr 459 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  <_  ( k ^ k ) )
34 nn0ge0 10738 . . . . . . . 8  |-  ( k  e.  NN0  ->  0  <_ 
k )
3534adantr 463 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
0  <_  k )
3626lep1d 10393 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
k  <_  ( k  +  1 ) )
37 leexp1a 12127 . . . . . . 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 1234 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k ^ k
)  <_  ( (
k  +  1 ) ^ k ) )
3924, 28, 32, 33, 38letrd 9650 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  k
)  <_  ( (
k  +  1 ) ^ k ) )
4030nngt0d 10496 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
0  <  ( k  +  1 ) )
41 lemul1 10311 . . . . . 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 1230 . . . . 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 12260 . . . . 5  |-  ( k  e.  NN0  ->  ( ! `
 ( k  +  1 ) )  =  ( ( ! `  k )  x.  (
k  +  1 ) ) )
4544adantr 463 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  (
k  +  1 ) )  =  ( ( ! `  k )  x.  ( k  +  1 ) ) )
4630nncnd 10468 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( k  +  1 )  e.  CC )
4746, 27expp1d 12213 . . . 4  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ( k  +  1 ) ^ (
k  +  1 ) )  =  ( ( ( k  +  1 ) ^ k )  x.  ( k  +  1 ) ) )
4843, 45, 473brtr4d 4397 . . 3  |-  ( ( k  e.  NN0  /\  ( ! `  k )  <_  ( k ^
k ) )  -> 
( ! `  (
k  +  1 ) )  <_  ( (
k  +  1 ) ^ ( k  +  1 ) ) )
4948ex 432 . 2  |-  ( k  e.  NN0  ->  ( ( ! `  k )  <_  ( k ^
k )  ->  ( ! `  ( k  +  1 ) )  <_  ( ( k  +  1 ) ^
( k  +  1 ) ) ) )
508, 12, 16, 20, 21, 49nn0ind 10874 1  |-  ( N  e.  NN0  ->  ( ! `
 N )  <_ 
( N ^ N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408    < clt 9539    <_ cle 9540   NNcn 10452   NN0cn0 10712   ^cexp 12069   !cfa 12255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-seq 12011  df-exp 12070  df-fac 12256
This theorem is referenced by:  logfacubnd  23613  pgrple2abl  33158
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