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Theorem log2ublem1 20778
Description: Lemma for log2ub 20781. The proof of log2ub 20781, which is simply the evaluation of log2tlbnd 20777 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 the 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 10126 . . . . . . . 8  |-  3  e.  NN
3 7nn0 10235 . . . . . . . 8  |-  7  e.  NN0
4 nnexpcl 11386 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
52, 3, 4mp2an 654 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
6 5nn 10128 . . . . . . . 8  |-  5  e.  NN
7 7nn 10130 . . . . . . . 8  |-  7  e.  NN
86, 7nnmulcli 10016 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
95, 8nnmulcli 10016 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
109nncni 10002 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
11 log2ublem1.3 . . . . . 6  |-  D  e. 
NN0
1211nn0cni 10225 . . . . 5  |-  D  e.  CC
13 log2ublem1.4 . . . . . 6  |-  E  e.  NN
1413nncni 10002 . . . . 5  |-  E  e.  CC
1513nnne0i 10026 . . . . 5  |-  E  =/=  0
1610, 12, 14, 15divassi 9762 . . . 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 10231 . . . . . . . . . 10  |-  3  e.  NN0
1918, 3nn0expcli 11399 . . . . . . . . 9  |-  ( 3 ^ 7 )  e. 
NN0
20 5nn0 10233 . . . . . . . . . 10  |-  5  e.  NN0
2120, 3nn0mulcli 10250 . . . . . . . . 9  |-  ( 5  x.  7 )  e. 
NN0
2219, 21nn0mulcli 10250 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e. 
NN0
2322, 11nn0mulcli 10250 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e. 
NN0
2423nn0rei 10224 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e.  RR
25 log2ublem1.6 . . . . . . 7  |-  F  e. 
NN0
2625nn0rei 10224 . . . . . 6  |-  F  e.  RR
2713nnrei 10001 . . . . . . 7  |-  E  e.  RR
2813nngt0i 10025 . . . . . . 7  |-  0  <  E
2927, 28pm3.2i 442 . . . . . 6  |-  ( E  e.  RR  /\  0  <  E )
30 ledivmul 9875 . . . . . 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 1279 . . . . 5  |-  ( ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D
)  /  E )  <_  F  <->  ( (
( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  <_ 
( E  x.  F
) )
3217, 31mpbir 201 . . . 4  |-  ( ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  D )  /  E )  <_  F
3316, 32eqbrtrri 4225 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  <_  F
349nnrei 10001 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
35 log2ublem1.2 . . . . 5  |-  A  e.  RR
3634, 35remulcli 9096 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  e.  RR
3711nn0rei 10224 . . . . . 6  |-  D  e.  RR
38 nndivre 10027 . . . . . 6  |-  ( ( D  e.  RR  /\  E  e.  NN )  ->  ( D  /  E
)  e.  RR )
3937, 13, 38mp2an 654 . . . . 5  |-  ( D  /  E )  e.  RR
4034, 39remulcli 9096 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  e.  RR
41 log2ublem1.5 . . . . 5  |-  B  e. 
NN0
4241nn0rei 10224 . . . 4  |-  B  e.  RR
4336, 40, 42, 26le2addi 9582 . . 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 654 . 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 6084 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  =  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( A  +  ( D  /  E ) ) )
4735recni 9094 . . . 4  |-  A  e.  CC
4839recni 9094 . . . 4  |-  ( D  /  E )  e.  CC
4910, 47, 48adddii 9092 . . 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 2456 . 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 4231 1  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  <_  G
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   RRcr 8981   0cc0 8982    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    / cdiv 9669   NNcn 9992   3c3 10042   5c5 10044   7c7 10046   NN0cn0 10213   ^cexp 11374
This theorem is referenced by:  log2ublem2  20779  log2ub  20781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-exp 11375
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