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Theorem log2ublem1 20169
Description: Lemma for log2ub 20172. The proof of log2ub 20172, which is simply the evaluation of log2tlbnd 20168 for  N  =  4, takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator  d (usually a large power of  10) and work with closest approximations of the form  n  /  d for some integer  n instead. It turns out that for our purposes it is sufficient to take  d  =  ( 3 ^ 7 )  x.  5  x.  7, which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)
Hypotheses
Ref Expression
log2ublem1.1  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  <_  B
log2ublem1.2  |-  A  e.  RR
log2ublem1.3  |-  D  e. 
NN0
log2ublem1.4  |-  E  e.  NN
log2ublem1.5  |-  B  e. 
NN0
log2ublem1.6  |-  F  e. 
NN0
log2ublem1.7  |-  C  =  ( A  +  ( D  /  E ) )
log2ublem1.8  |-  ( B  +  F )  =  G
log2ublem1.9  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  <_ 
( E  x.  F
)
Assertion
Ref Expression
log2ublem1  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  <_  G

Proof of Theorem log2ublem1
StepHypRef Expression
1 log2ublem1.1 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  <_  B
2 3nn 9810 . . . . . . . 8  |-  3  e.  NN
3 7nn0 9919 . . . . . . . 8  |-  7  e.  NN0
4 nnexpcl 11047 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
52, 3, 4mp2an 656 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
6 5nn 9812 . . . . . . . 8  |-  5  e.  NN
7 7nn 9814 . . . . . . . 8  |-  7  e.  NN
86, 7nnmulcli 9703 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
95, 8nnmulcli 9703 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
109nncni 9689 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
11 log2ublem1.3 . . . . . 6  |-  D  e. 
NN0
1211nn0cni 9909 . . . . 5  |-  D  e.  CC
13 log2ublem1.4 . . . . . 6  |-  E  e.  NN
1413nncni 9689 . . . . 5  |-  E  e.  CC
1513nnne0i 9713 . . . . 5  |-  E  =/=  0
1610, 12, 14, 15divassi 9449 . . . 4  |-  ( ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  D )  /  E )  =  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )
17 log2ublem1.9 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  <_ 
( E  x.  F
)
18 3nn0 9915 . . . . . . . . . 10  |-  3  e.  NN0
1918, 3nn0expcli 11060 . . . . . . . . 9  |-  ( 3 ^ 7 )  e. 
NN0
20 5nn0 9917 . . . . . . . . . 10  |-  5  e.  NN0
2120, 3nn0mulcli 9934 . . . . . . . . 9  |-  ( 5  x.  7 )  e. 
NN0
2219, 21nn0mulcli 9934 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e. 
NN0
2322, 11nn0mulcli 9934 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e. 
NN0
2423nn0rei 9908 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e.  RR
25 log2ublem1.6 . . . . . . 7  |-  F  e. 
NN0
2625nn0rei 9908 . . . . . 6  |-  F  e.  RR
2713nnrei 9688 . . . . . . 7  |-  E  e.  RR
2813nngt0i 9712 . . . . . . 7  |-  0  <  E
2927, 28pm3.2i 443 . . . . . 6  |-  ( E  e.  RR  /\  0  <  E )
30 ledivmul 9562 . . . . . 6  |-  ( ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D
)  e.  RR  /\  F  e.  RR  /\  ( E  e.  RR  /\  0  <  E ) )  -> 
( ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  /  E )  <_  F  <->  ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  D )  <_  ( E  x.  F ) ) )
3124, 26, 29, 30mp3an 1282 . . . . 5  |-  ( ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D
)  /  E )  <_  F  <->  ( (
( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  <_ 
( E  x.  F
) )
3217, 31mpbir 202 . . . 4  |-  ( ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  D )  /  E )  <_  F
3316, 32eqbrtrri 3984 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  <_  F
349nnrei 9688 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
35 log2ublem1.2 . . . . 5  |-  A  e.  RR
3634, 35remulcli 8784 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  e.  RR
3711nn0rei 9908 . . . . . 6  |-  D  e.  RR
38 nndivre 9714 . . . . . 6  |-  ( ( D  e.  RR  /\  E  e.  NN )  ->  ( D  /  E
)  e.  RR )
3937, 13, 38mp2an 656 . . . . 5  |-  ( D  /  E )  e.  RR
4034, 39remulcli 8784 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  e.  RR
41 log2ublem1.5 . . . . 5  |-  B  e. 
NN0
4241nn0rei 9908 . . . 4  |-  B  e.  RR
4336, 40, 42, 26le2addi 9269 . . 3  |-  ( ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A
)  <_  B  /\  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  <_  F )  -> 
( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  +  ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  ( D  /  E ) ) )  <_  ( B  +  F ) )
441, 33, 43mp2an 656 . 2  |-  ( ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  A )  +  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E
) ) )  <_ 
( B  +  F
)
45 log2ublem1.7 . . . 4  |-  C  =  ( A  +  ( D  /  E ) )
4645oveq2i 5768 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  =  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( A  +  ( D  /  E ) ) )
4735recni 8782 . . . 4  |-  A  e.  CC
4839recni 8782 . . . 4  |-  ( D  /  E )  e.  CC
4910, 47, 48adddii 8780 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( A  +  ( D  /  E
) ) )  =  ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  +  ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  ( D  /  E ) ) )
5046, 49eqtr2i 2277 . 2  |-  ( ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  A )  +  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E
) ) )  =  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C
)
51 log2ublem1.8 . 2  |-  ( B  +  F )  =  G
5244, 50, 513brtr3i 3990 1  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  <_  G
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3963  (class class class)co 5757   RRcr 8669   0cc0 8670    + caddc 8673    x. cmul 8675    < clt 8800    <_ cle 8801    / cdiv 9356   NNcn 9679   3c3 9729   5c5 9731   7c7 9733   NN0cn0 9897   ^cexp 11035
This theorem is referenced by:  log2ublem2  20170  log2ub  20172
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-n0 9898  df-z 9957  df-uz 10163  df-seq 10978  df-exp 11036
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