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Theorem log2ublem1 20074
 Description: Lemma for log2ub 20077. The proof of log2ub 20077, which is simply the evaluation of log2tlbnd 20073 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 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 9757 . . . . . . . 8
3 7nn0 9866 . . . . . . . 8
4 nnexpcl 10994 . . . . . . . 8
52, 3, 4mp2an 656 . . . . . . 7
6 5nn 9759 . . . . . . . 8
7 7nn 9761 . . . . . . . 8
86, 7nnmulcli 9650 . . . . . . 7
95, 8nnmulcli 9650 . . . . . 6
109nncni 9636 . . . . 5
11 log2ublem1.3 . . . . . 6
1211nn0cni 9856 . . . . 5
13 log2ublem1.4 . . . . . 6
1413nncni 9636 . . . . 5
1513nnne0i 9660 . . . . 5
1610, 12, 14, 15divassi 9396 . . . 4
17 log2ublem1.9 . . . . 5
18 3nn0 9862 . . . . . . . . . 10
1918, 3nn0expcli 11007 . . . . . . . . 9
20 5nn0 9864 . . . . . . . . . 10
2120, 3nn0mulcli 9881 . . . . . . . . 9
2219, 21nn0mulcli 9881 . . . . . . . 8
2322, 11nn0mulcli 9881 . . . . . . 7
2423nn0rei 9855 . . . . . 6
25 log2ublem1.6 . . . . . . 7
2625nn0rei 9855 . . . . . 6
2713nnrei 9635 . . . . . . 7
2813nngt0i 9659 . . . . . . 7
2927, 28pm3.2i 443 . . . . . 6
30 ledivmul 9509 . . . . . 6
3124, 26, 29, 30mp3an 1282 . . . . 5
3217, 31mpbir 202 . . . 4
3316, 32eqbrtrri 3941 . . 3
349nnrei 9635 . . . . 5
35 log2ublem1.2 . . . . 5
3634, 35remulcli 8731 . . . 4
3711nn0rei 9855 . . . . . 6
38 nndivre 9661 . . . . . 6
3937, 13, 38mp2an 656 . . . . 5
4034, 39remulcli 8731 . . . 4
41 log2ublem1.5 . . . . 5
4241nn0rei 9855 . . . 4
4336, 40, 42, 26le2addi 9216 . . 3
441, 33, 43mp2an 656 . 2
45 log2ublem1.7 . . . 4
4645oveq2i 5721 . . 3
4735recni 8729 . . . 4
4839recni 8729 . . . 4
4910, 47, 48adddii 8727 . . 3
5046, 49eqtr2i 2274 . 2
51 log2ublem1.8 . 2
5244, 50, 513brtr3i 3947 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1619   wcel 1621   class class class wbr 3920  (class class class)co 5710  cr 8616  cc0 8617   caddc 8620   cmul 8622   clt 8747   cle 8748   cdiv 9303  cn 9626  c3 9676  c5 9678  c7 9680  cn0 9844  cexp 10982 This theorem is referenced by:  log2ublem2  20075  log2ub  20077 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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694 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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-n0 9845  df-z 9904  df-uz 10110  df-seq 10925  df-exp 10983
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