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Theorem faclbnd4lem3 11308
Description: Lemma for faclbnd4 11310. The  N  =  0 case. (Contributed by NM, 23-Dec-2005.)
Assertion
Ref Expression
faclbnd4lem3  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( ( N ^ K )  x.  ( M ^ N
) )  <_  (
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  x.  ( ! `  N ) ) )

Proof of Theorem faclbnd4lem3
StepHypRef Expression
1 elnn0 9967 . . . . 5  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 0exp 11137 . . . . . . . 8  |-  ( K  e.  NN  ->  (
0 ^ K )  =  0 )
32adantl 452 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( 0 ^ K
)  =  0 )
4 nnnn0 9972 . . . . . . . . 9  |-  ( K  e.  NN  ->  K  e.  NN0 )
5 2nn0 9982 . . . . . . . . . . . 12  |-  2  e.  NN0
6 nn0expcl 11117 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  2  e.  NN0 )  -> 
( K ^ 2 )  e.  NN0 )
75, 6mpan2 652 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K ^ 2 )  e. 
NN0 )
8 nn0expcl 11117 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN0  /\  ( K ^ 2 )  e.  NN0 )  -> 
( 2 ^ ( K ^ 2 ) )  e.  NN0 )
95, 7, 8sylancr 644 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  ( 2 ^ ( K ^
2 ) )  e. 
NN0 )
109adantl 452 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( 2 ^ ( K ^ 2 ) )  e.  NN0 )
11 nn0addcl 9999 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( M  +  K
)  e.  NN0 )
12 nn0expcl 11117 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  ( M  +  K
)  e.  NN0 )  ->  ( M ^ ( M  +  K )
)  e.  NN0 )
1311, 12syldan 456 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( M ^ ( M  +  K )
)  e.  NN0 )
1410, 13nn0mulcld 10023 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  NN0 )
154, 14sylan2 460 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  NN0 )
1615nn0ge0d 10021 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  0  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
173, 16eqbrtrd 4043 . . . . . 6  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
18 1nn 9757 . . . . . . . . . 10  |-  1  e.  NN
19 elnn0 9967 . . . . . . . . . . 11  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
20 nnnn0 9972 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  ->  M  e.  NN0 )
21 0nn0 9980 . . . . . . . . . . . . . 14  |-  0  e.  NN0
22 nn0addcl 9999 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN0  /\  0  e.  NN0 )  -> 
( M  +  0 )  e.  NN0 )
2320, 21, 22sylancl 643 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  ( M  +  0 )  e.  NN0 )
24 nnexpcl 11116 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  ( M  +  0
)  e.  NN0 )  ->  ( M ^ ( M  +  0 ) )  e.  NN )
2523, 24mpdan 649 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
26 id 19 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  M  =  0 )
27 oveq1 5865 . . . . . . . . . . . . . . . 16  |-  ( M  =  0  ->  ( M  +  0 )  =  ( 0  +  0 ) )
28 00id 8987 . . . . . . . . . . . . . . . 16  |-  ( 0  +  0 )  =  0
2927, 28syl6eq 2331 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  ( M  +  0 )  =  0 )
3026, 29oveq12d 5876 . . . . . . . . . . . . . 14  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  =  ( 0 ^ 0 ) )
31 0cn 8831 . . . . . . . . . . . . . . 15  |-  0  e.  CC
32 exp0 11108 . . . . . . . . . . . . . . 15  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
3331, 32ax-mp 8 . . . . . . . . . . . . . 14  |-  ( 0 ^ 0 )  =  1
3430, 33syl6eq 2331 . . . . . . . . . . . . 13  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  =  1 )
3534, 18syl6eqel 2371 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
3625, 35jaoi 368 . . . . . . . . . . 11  |-  ( ( M  e.  NN  \/  M  =  0 )  ->  ( M ^
( M  +  0 ) )  e.  NN )
3719, 36sylbi 187 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
38 nnmulcl 9769 . . . . . . . . . 10  |-  ( ( 1  e.  NN  /\  ( M ^ ( M  +  0 ) )  e.  NN )  -> 
( 1  x.  ( M ^ ( M  + 
0 ) ) )  e.  NN )
3918, 37, 38sylancr 644 . . . . . . . . 9  |-  ( M  e.  NN0  ->  ( 1  x.  ( M ^
( M  +  0 ) ) )  e.  NN )
4039nnge1d 9788 . . . . . . . 8  |-  ( M  e.  NN0  ->  1  <_ 
( 1  x.  ( M ^ ( M  + 
0 ) ) ) )
4140adantr 451 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  1  <_  (
1  x.  ( M ^ ( M  + 
0 ) ) ) )
42 oveq2 5866 . . . . . . . . . 10  |-  ( K  =  0  ->  (
0 ^ K )  =  ( 0 ^ 0 ) )
4342, 33syl6eq 2331 . . . . . . . . 9  |-  ( K  =  0  ->  (
0 ^ K )  =  1 )
44 sq0i 11196 . . . . . . . . . . . 12  |-  ( K  =  0  ->  ( K ^ 2 )  =  0 )
4544oveq2d 5874 . . . . . . . . . . 11  |-  ( K  =  0  ->  (
2 ^ ( K ^ 2 ) )  =  ( 2 ^ 0 ) )
46 2cn 9816 . . . . . . . . . . . 12  |-  2  e.  CC
47 exp0 11108 . . . . . . . . . . . 12  |-  ( 2  e.  CC  ->  (
2 ^ 0 )  =  1 )
4846, 47ax-mp 8 . . . . . . . . . . 11  |-  ( 2 ^ 0 )  =  1
4945, 48syl6eq 2331 . . . . . . . . . 10  |-  ( K  =  0  ->  (
2 ^ ( K ^ 2 ) )  =  1 )
50 oveq2 5866 . . . . . . . . . . 11  |-  ( K  =  0  ->  ( M  +  K )  =  ( M  + 
0 ) )
5150oveq2d 5874 . . . . . . . . . 10  |-  ( K  =  0  ->  ( M ^ ( M  +  K ) )  =  ( M ^ ( M  +  0 ) ) )
5249, 51oveq12d 5876 . . . . . . . . 9  |-  ( K  =  0  ->  (
( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  =  ( 1  x.  ( M ^ ( M  + 
0 ) ) ) )
5343, 52breq12d 4036 . . . . . . . 8  |-  ( K  =  0  ->  (
( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  <->  1  <_  ( 1  x.  ( M ^ ( M  + 
0 ) ) ) ) )
5453adantl 452 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  ( ( 0 ^ K )  <_ 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  <->  1  <_  ( 1  x.  ( M ^
( M  +  0 ) ) ) ) )
5541, 54mpbird 223 . . . . . 6  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  ( 0 ^ K )  <_  (
( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
5617, 55jaodan 760 . . . . 5  |-  ( ( M  e.  NN0  /\  ( K  e.  NN  \/  K  =  0
) )  ->  (
0 ^ K )  <_  ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) ) )
571, 56sylan2b 461 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
58 nn0cn 9975 . . . . . . 7  |-  ( M  e.  NN0  ->  M  e.  CC )
5958exp0d 11239 . . . . . 6  |-  ( M  e.  NN0  ->  ( M ^ 0 )  =  1 )
6059oveq2d 5874 . . . . 5  |-  ( M  e.  NN0  ->  ( ( 0 ^ K )  x.  ( M ^
0 ) )  =  ( ( 0 ^ K )  x.  1 ) )
61 nn0expcl 11117 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  K  e.  NN0 )  -> 
( 0 ^ K
)  e.  NN0 )
6221, 61mpan 651 . . . . . . 7  |-  ( K  e.  NN0  ->  ( 0 ^ K )  e. 
NN0 )
6362nn0cnd 10020 . . . . . 6  |-  ( K  e.  NN0  ->  ( 0 ^ K )  e.  CC )
6463mulid1d 8852 . . . . 5  |-  ( K  e.  NN0  ->  ( ( 0 ^ K )  x.  1 )  =  ( 0 ^ K
) )
6560, 64sylan9eq 2335 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 0 ^ K )  x.  ( M ^ 0 ) )  =  ( 0 ^ K ) )
6614nn0cnd 10020 . . . . 5  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  CC )
6766mulid1d 8852 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 )  =  ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
6857, 65, 673brtr4d 4053 . . 3  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 0 ^ K )  x.  ( M ^ 0 ) )  <_  ( ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  x.  1 ) )
6968adantr 451 . 2  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( (
0 ^ K )  x.  ( M ^
0 ) )  <_ 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) )
70 oveq1 5865 . . . . 5  |-  ( N  =  0  ->  ( N ^ K )  =  ( 0 ^ K
) )
71 oveq2 5866 . . . . 5  |-  ( N  =  0  ->  ( M ^ N )  =  ( M ^ 0 ) )
7270, 71oveq12d 5876 . . . 4  |-  ( N  =  0  ->  (
( N ^ K
)  x.  ( M ^ N ) )  =  ( ( 0 ^ K )  x.  ( M ^ 0 ) ) )
73 fveq2 5525 . . . . . 6  |-  ( N  =  0  ->  ( ! `  N )  =  ( ! ` 
0 ) )
74 fac0 11291 . . . . . 6  |-  ( ! `
 0 )  =  1
7573, 74syl6eq 2331 . . . . 5  |-  ( N  =  0  ->  ( ! `  N )  =  1 )
7675oveq2d 5874 . . . 4  |-  ( N  =  0  ->  (
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  x.  ( ! `  N ) )  =  ( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) )
7772, 76breq12d 4036 . . 3  |-  ( N  =  0  ->  (
( ( N ^ K )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  x.  ( ! `  N
) )  <->  ( (
0 ^ K )  x.  ( M ^
0 ) )  <_ 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) ) )
7877adantl 452 . 2  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( (
( N ^ K
)  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  x.  ( ! `  N
) )  <->  ( (
0 ^ K )  x.  ( M ^
0 ) )  <_ 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) ) )
7969, 78mpbird 223 1  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( ( N ^ K )  x.  ( M ^ N
) )  <_  (
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  x.  ( ! `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868   NNcn 9746   2c2 9795   NN0cn0 9965   ^cexp 11104   !cfa 11288
This theorem is referenced by:  faclbnd4lem4  11309  faclbnd4  11310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105  df-fac 11289
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