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Theorem log2ublem1 23149
Description: Lemma for log2ub 23152. The proof of log2ub 23152, which is simply the evaluation of log2tlbnd 23148 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 10700 . . . . . . . 8  |-  3  e.  NN
3 7nn0 10823 . . . . . . . 8  |-  7  e.  NN0
4 nnexpcl 12158 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
52, 3, 4mp2an 672 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
6 5nn 10702 . . . . . . . 8  |-  5  e.  NN
7 7nn 10704 . . . . . . . 8  |-  7  e.  NN
86, 7nnmulcli 10566 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
95, 8nnmulcli 10566 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
109nncni 10552 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
11 log2ublem1.3 . . . . . 6  |-  D  e. 
NN0
1211nn0cni 10813 . . . . 5  |-  D  e.  CC
13 log2ublem1.4 . . . . . 6  |-  E  e.  NN
1413nncni 10552 . . . . 5  |-  E  e.  CC
1513nnne0i 10576 . . . . 5  |-  E  =/=  0
1610, 12, 14, 15divassi 10306 . . . 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 10819 . . . . . . . . . 10  |-  3  e.  NN0
1918, 3nn0expcli 12171 . . . . . . . . 9  |-  ( 3 ^ 7 )  e. 
NN0
20 5nn0 10821 . . . . . . . . . 10  |-  5  e.  NN0
2120, 3nn0mulcli 10840 . . . . . . . . 9  |-  ( 5  x.  7 )  e. 
NN0
2219, 21nn0mulcli 10840 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e. 
NN0
2322, 11nn0mulcli 10840 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e. 
NN0
2423nn0rei 10812 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e.  RR
25 log2ublem1.6 . . . . . . 7  |-  F  e. 
NN0
2625nn0rei 10812 . . . . . 6  |-  F  e.  RR
2713nnrei 10551 . . . . . . 7  |-  E  e.  RR
2813nngt0i 10575 . . . . . . 7  |-  0  <  E
2927, 28pm3.2i 455 . . . . . 6  |-  ( E  e.  RR  /\  0  <  E )
30 ledivmul 10424 . . . . . 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 1325 . . . . 5  |-  ( ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D
)  /  E )  <_  F  <->  ( (
( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  <_ 
( E  x.  F
) )
3217, 31mpbir 209 . . . 4  |-  ( ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  D )  /  E )  <_  F
3316, 32eqbrtrri 4458 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  <_  F
349nnrei 10551 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
35 log2ublem1.2 . . . . 5  |-  A  e.  RR
3634, 35remulcli 9613 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  e.  RR
3711nn0rei 10812 . . . . . 6  |-  D  e.  RR
38 nndivre 10577 . . . . . 6  |-  ( ( D  e.  RR  /\  E  e.  NN )  ->  ( D  /  E
)  e.  RR )
3937, 13, 38mp2an 672 . . . . 5  |-  ( D  /  E )  e.  RR
4034, 39remulcli 9613 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  e.  RR
41 log2ublem1.5 . . . . 5  |-  B  e. 
NN0
4241nn0rei 10812 . . . 4  |-  B  e.  RR
4336, 40, 42, 26le2addi 10122 . . 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 672 . 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 6292 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  =  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( A  +  ( D  /  E ) ) )
4735recni 9611 . . . 4  |-  A  e.  CC
4839recni 9611 . . . 4  |-  ( D  /  E )  e.  CC
4910, 47, 48adddii 9609 . . 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 2473 . 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 4464 1  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  <_  G
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   class class class wbr 4437  (class class class)co 6281   RRcr 9494   0cc0 9495    + caddc 9498    x. cmul 9500    < clt 9631    <_ cle 9632    / cdiv 10212   NNcn 10542   3c3 10592   5c5 10594   7c7 10596   NN0cn0 10801   ^cexp 12145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-n0 10802  df-z 10871  df-uz 11091  df-seq 12087  df-exp 12146
This theorem is referenced by:  log2ublem2  23150  log2ub  23152
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