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Theorem faclbnd4lem3 12328
Description: Lemma for faclbnd4 12330. 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 10786 . . . . 5  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 0exp 12156 . . . . . . . 8  |-  ( K  e.  NN  ->  (
0 ^ K )  =  0 )
32adantl 466 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( 0 ^ K
)  =  0 )
4 nnnn0 10791 . . . . . . . . 9  |-  ( K  e.  NN  ->  K  e.  NN0 )
5 2nn0 10801 . . . . . . . . . . . 12  |-  2  e.  NN0
6 nn0expcl 12136 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  2  e.  NN0 )  -> 
( K ^ 2 )  e.  NN0 )
75, 6mpan2 671 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K ^ 2 )  e. 
NN0 )
8 nn0expcl 12136 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN0  /\  ( K ^ 2 )  e.  NN0 )  -> 
( 2 ^ ( K ^ 2 ) )  e.  NN0 )
95, 7, 8sylancr 663 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  ( 2 ^ ( K ^
2 ) )  e. 
NN0 )
109adantl 466 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( 2 ^ ( K ^ 2 ) )  e.  NN0 )
11 nn0addcl 10820 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( M  +  K
)  e.  NN0 )
12 nn0expcl 12136 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  ( M  +  K
)  e.  NN0 )  ->  ( M ^ ( M  +  K )
)  e.  NN0 )
1311, 12syldan 470 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( M ^ ( M  +  K )
)  e.  NN0 )
1410, 13nn0mulcld 10846 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  NN0 )
154, 14sylan2 474 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  NN0 )
1615nn0ge0d 10844 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  0  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
173, 16eqbrtrd 4460 . . . . . 6  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
18 1nn 10536 . . . . . . . . . 10  |-  1  e.  NN
19 elnn0 10786 . . . . . . . . . . 11  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
20 nnnn0 10791 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  ->  M  e.  NN0 )
21 0nn0 10799 . . . . . . . . . . . . . 14  |-  0  e.  NN0
22 nn0addcl 10820 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN0  /\  0  e.  NN0 )  -> 
( M  +  0 )  e.  NN0 )
2320, 21, 22sylancl 662 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  ( M  +  0 )  e.  NN0 )
24 nnexpcl 12135 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  ( M  +  0
)  e.  NN0 )  ->  ( M ^ ( M  +  0 ) )  e.  NN )
2523, 24mpdan 668 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
26 id 22 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  M  =  0 )
27 oveq1 6282 . . . . . . . . . . . . . . . 16  |-  ( M  =  0  ->  ( M  +  0 )  =  ( 0  +  0 ) )
28 00id 9743 . . . . . . . . . . . . . . . 16  |-  ( 0  +  0 )  =  0
2927, 28syl6eq 2517 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  ( M  +  0 )  =  0 )
3026, 29oveq12d 6293 . . . . . . . . . . . . . 14  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  =  ( 0 ^ 0 ) )
31 0exp0e1 12127 . . . . . . . . . . . . . 14  |-  ( 0 ^ 0 )  =  1
3230, 31syl6eq 2517 . . . . . . . . . . . . 13  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  =  1 )
3332, 18syl6eqel 2556 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
3425, 33jaoi 379 . . . . . . . . . . 11  |-  ( ( M  e.  NN  \/  M  =  0 )  ->  ( M ^
( M  +  0 ) )  e.  NN )
3519, 34sylbi 195 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
36 nnmulcl 10548 . . . . . . . . . 10  |-  ( ( 1  e.  NN  /\  ( M ^ ( M  +  0 ) )  e.  NN )  -> 
( 1  x.  ( M ^ ( M  + 
0 ) ) )  e.  NN )
3718, 35, 36sylancr 663 . . . . . . . . 9  |-  ( M  e.  NN0  ->  ( 1  x.  ( M ^
( M  +  0 ) ) )  e.  NN )
3837nnge1d 10567 . . . . . . . 8  |-  ( M  e.  NN0  ->  1  <_ 
( 1  x.  ( M ^ ( M  + 
0 ) ) ) )
3938adantr 465 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  1  <_  (
1  x.  ( M ^ ( M  + 
0 ) ) ) )
40 oveq2 6283 . . . . . . . . . 10  |-  ( K  =  0  ->  (
0 ^ K )  =  ( 0 ^ 0 ) )
4140, 31syl6eq 2517 . . . . . . . . 9  |-  ( K  =  0  ->  (
0 ^ K )  =  1 )
42 sq0i 12215 . . . . . . . . . . . 12  |-  ( K  =  0  ->  ( K ^ 2 )  =  0 )
4342oveq2d 6291 . . . . . . . . . . 11  |-  ( K  =  0  ->  (
2 ^ ( K ^ 2 ) )  =  ( 2 ^ 0 ) )
44 2cn 10595 . . . . . . . . . . . 12  |-  2  e.  CC
45 exp0 12126 . . . . . . . . . . . 12  |-  ( 2  e.  CC  ->  (
2 ^ 0 )  =  1 )
4644, 45ax-mp 5 . . . . . . . . . . 11  |-  ( 2 ^ 0 )  =  1
4743, 46syl6eq 2517 . . . . . . . . . 10  |-  ( K  =  0  ->  (
2 ^ ( K ^ 2 ) )  =  1 )
48 oveq2 6283 . . . . . . . . . . 11  |-  ( K  =  0  ->  ( M  +  K )  =  ( M  + 
0 ) )
4948oveq2d 6291 . . . . . . . . . 10  |-  ( K  =  0  ->  ( M ^ ( M  +  K ) )  =  ( M ^ ( M  +  0 ) ) )
5047, 49oveq12d 6293 . . . . . . . . 9  |-  ( K  =  0  ->  (
( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  =  ( 1  x.  ( M ^ ( M  + 
0 ) ) ) )
5141, 50breq12d 4453 . . . . . . . 8  |-  ( K  =  0  ->  (
( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  <->  1  <_  ( 1  x.  ( M ^ ( M  + 
0 ) ) ) ) )
5251adantl 466 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  ( ( 0 ^ K )  <_ 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  <->  1  <_  ( 1  x.  ( M ^
( M  +  0 ) ) ) ) )
5339, 52mpbird 232 . . . . . 6  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  ( 0 ^ K )  <_  (
( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
5417, 53jaodan 783 . . . . 5  |-  ( ( M  e.  NN0  /\  ( K  e.  NN  \/  K  =  0
) )  ->  (
0 ^ K )  <_  ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) ) )
551, 54sylan2b 475 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
56 nn0cn 10794 . . . . . . 7  |-  ( M  e.  NN0  ->  M  e.  CC )
5756exp0d 12259 . . . . . 6  |-  ( M  e.  NN0  ->  ( M ^ 0 )  =  1 )
5857oveq2d 6291 . . . . 5  |-  ( M  e.  NN0  ->  ( ( 0 ^ K )  x.  ( M ^
0 ) )  =  ( ( 0 ^ K )  x.  1 ) )
59 nn0expcl 12136 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  K  e.  NN0 )  -> 
( 0 ^ K
)  e.  NN0 )
6021, 59mpan 670 . . . . . . 7  |-  ( K  e.  NN0  ->  ( 0 ^ K )  e. 
NN0 )
6160nn0cnd 10843 . . . . . 6  |-  ( K  e.  NN0  ->  ( 0 ^ K )  e.  CC )
6261mulid1d 9602 . . . . 5  |-  ( K  e.  NN0  ->  ( ( 0 ^ K )  x.  1 )  =  ( 0 ^ K
) )
6358, 62sylan9eq 2521 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 0 ^ K )  x.  ( M ^ 0 ) )  =  ( 0 ^ K ) )
6414nn0cnd 10843 . . . . 5  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  CC )
6564mulid1d 9602 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 )  =  ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
6655, 63, 653brtr4d 4470 . . 3  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 0 ^ K )  x.  ( M ^ 0 ) )  <_  ( ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  x.  1 ) )
6766adantr 465 . 2  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( (
0 ^ K )  x.  ( M ^
0 ) )  <_ 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) )
68 oveq1 6282 . . . . 5  |-  ( N  =  0  ->  ( N ^ K )  =  ( 0 ^ K
) )
69 oveq2 6283 . . . . 5  |-  ( N  =  0  ->  ( M ^ N )  =  ( M ^ 0 ) )
7068, 69oveq12d 6293 . . . 4  |-  ( N  =  0  ->  (
( N ^ K
)  x.  ( M ^ N ) )  =  ( ( 0 ^ K )  x.  ( M ^ 0 ) ) )
71 fveq2 5857 . . . . . 6  |-  ( N  =  0  ->  ( ! `  N )  =  ( ! ` 
0 ) )
72 fac0 12311 . . . . . 6  |-  ( ! `
 0 )  =  1
7371, 72syl6eq 2517 . . . . 5  |-  ( N  =  0  ->  ( ! `  N )  =  1 )
7473oveq2d 6291 . . . 4  |-  ( N  =  0  ->  (
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  x.  ( ! `  N ) )  =  ( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) )
7570, 74breq12d 4453 . . 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 ) ) )
7675adantl 466 . 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 ) ) )
7767, 76mpbird 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    <_ cle 9618   NNcn 10525   2c2 10574   NN0cn0 10784   ^cexp 12122   !cfa 12308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-seq 12064  df-exp 12123  df-fac 12309
This theorem is referenced by:  faclbnd4lem4  12329  faclbnd4  12330
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