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Theorem log2ublem1 20205
Description: Lemma for log2ub 20208. The proof of log2ub 20208, which is simply the evaluation of log2tlbnd 20204 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 9846 . . . . . . . 8  |-  3  e.  NN
3 7nn0 9955 . . . . . . . 8  |-  7  e.  NN0
4 nnexpcl 11083 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
52, 3, 4mp2an 656 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
6 5nn 9848 . . . . . . . 8  |-  5  e.  NN
7 7nn 9850 . . . . . . . 8  |-  7  e.  NN
86, 7nnmulcli 9738 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
95, 8nnmulcli 9738 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
109nncni 9724 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
11 log2ublem1.3 . . . . . 6  |-  D  e. 
NN0
1211nn0cni 9945 . . . . 5  |-  D  e.  CC
13 log2ublem1.4 . . . . . 6  |-  E  e.  NN
1413nncni 9724 . . . . 5  |-  E  e.  CC
1513nnne0i 9748 . . . . 5  |-  E  =/=  0
1610, 12, 14, 15divassi 9484 . . . 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 9951 . . . . . . . . . 10  |-  3  e.  NN0
1918, 3nn0expcli 11096 . . . . . . . . 9  |-  ( 3 ^ 7 )  e. 
NN0
20 5nn0 9953 . . . . . . . . . 10  |-  5  e.  NN0
2120, 3nn0mulcli 9970 . . . . . . . . 9  |-  ( 5  x.  7 )  e. 
NN0
2219, 21nn0mulcli 9970 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e. 
NN0
2322, 11nn0mulcli 9970 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e. 
NN0
2423nn0rei 9944 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e.  RR
25 log2ublem1.6 . . . . . . 7  |-  F  e. 
NN0
2625nn0rei 9944 . . . . . 6  |-  F  e.  RR
2713nnrei 9723 . . . . . . 7  |-  E  e.  RR
2813nngt0i 9747 . . . . . . 7  |-  0  <  E
2927, 28pm3.2i 443 . . . . . 6  |-  ( E  e.  RR  /\  0  <  E )
30 ledivmul 9597 . . . . . 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 4018 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  <_  F
349nnrei 9723 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
35 log2ublem1.2 . . . . 5  |-  A  e.  RR
3634, 35remulcli 8819 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  e.  RR
3711nn0rei 9944 . . . . . 6  |-  D  e.  RR
38 nndivre 9749 . . . . . 6  |-  ( ( D  e.  RR  /\  E  e.  NN )  ->  ( D  /  E
)  e.  RR )
3937, 13, 38mp2an 656 . . . . 5  |-  ( D  /  E )  e.  RR
4034, 39remulcli 8819 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  e.  RR
41 log2ublem1.5 . . . . 5  |-  B  e. 
NN0
4241nn0rei 9944 . . . 4  |-  B  e.  RR
4336, 40, 42, 26le2addi 9304 . . 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 5803 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  =  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( A  +  ( D  /  E ) ) )
4735recni 8817 . . . 4  |-  A  e.  CC
4839recni 8817 . . . 4  |-  ( D  /  E )  e.  CC
4910, 47, 48adddii 8815 . . 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 2279 . 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 4024 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 3997  (class class class)co 5792   RRcr 8704   0cc0 8705    + caddc 8708    x. cmul 8710    < clt 8835    <_ cle 8836    / cdiv 9391   NNcn 9714   3c3 9764   5c5 9766   7c7 9768   NN0cn0 9933   ^cexp 11071
This theorem is referenced by:  log2ublem2  20206  log2ub  20208
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-n0 9934  df-z 9993  df-uz 10199  df-seq 11014  df-exp 11072
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