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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 , 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 (usually a large power of ) and work with the closest approximations of the form for some integer instead. It turns out that for our purposes it is sufficient to take , 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
log2ublem1.2
log2ublem1.3
log2ublem1.4
log2ublem1.5
log2ublem1.6
log2ublem1.7
log2ublem1.8
log2ublem1.9
Assertion
Ref Expression
log2ublem1

Proof of Theorem log2ublem1
StepHypRef Expression
1 log2ublem1.1 . . 3
2 3nn 10126 . . . . . . . 8
3 7nn0 10235 . . . . . . . 8
4 nnexpcl 11386 . . . . . . . 8
52, 3, 4mp2an 654 . . . . . . 7
6 5nn 10128 . . . . . . . 8
7 7nn 10130 . . . . . . . 8
86, 7nnmulcli 10016 . . . . . . 7
95, 8nnmulcli 10016 . . . . . 6
109nncni 10002 . . . . 5
11 log2ublem1.3 . . . . . 6
1211nn0cni 10225 . . . . 5
13 log2ublem1.4 . . . . . 6
1413nncni 10002 . . . . 5
1513nnne0i 10026 . . . . 5
1610, 12, 14, 15divassi 9762 . . . 4
17 log2ublem1.9 . . . . 5
18 3nn0 10231 . . . . . . . . . 10
1918, 3nn0expcli 11399 . . . . . . . . 9
20 5nn0 10233 . . . . . . . . . 10
2120, 3nn0mulcli 10250 . . . . . . . . 9
2219, 21nn0mulcli 10250 . . . . . . . 8
2322, 11nn0mulcli 10250 . . . . . . 7
2423nn0rei 10224 . . . . . 6
25 log2ublem1.6 . . . . . . 7
2625nn0rei 10224 . . . . . 6
2713nnrei 10001 . . . . . . 7
2813nngt0i 10025 . . . . . . 7
2927, 28pm3.2i 442 . . . . . 6
30 ledivmul 9875 . . . . . 6
3124, 26, 29, 30mp3an 1279 . . . . 5
3217, 31mpbir 201 . . . 4
3316, 32eqbrtrri 4225 . . 3
349nnrei 10001 . . . . 5
35 log2ublem1.2 . . . . 5
3634, 35remulcli 9096 . . . 4
3711nn0rei 10224 . . . . . 6
38 nndivre 10027 . . . . . 6
3937, 13, 38mp2an 654 . . . . 5
4034, 39remulcli 9096 . . . 4
41 log2ublem1.5 . . . . 5
4241nn0rei 10224 . . . 4
4336, 40, 42, 26le2addi 9582 . . 3
441, 33, 43mp2an 654 . 2
45 log2ublem1.7 . . . 4
4645oveq2i 6084 . . 3
4735recni 9094 . . . 4
4839recni 9094 . . . 4
4910, 47, 48adddii 9092 . . 3
5046, 49eqtr2i 2456 . 2
51 log2ublem1.8 . 2
5244, 50, 513brtr3i 4231 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725   class class class wbr 4204  (class class class)co 6073  cr 8981  cc0 8982   caddc 8985   cmul 8987   clt 9112   cle 9113   cdiv 9669  cn 9992  c3 10042  c5 10044  c7 10046  cn0 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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