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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  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 9757 . . . . . . . 8  |-  3  e.  NN
3 7nn0 9866 . . . . . . . 8  |-  7  e.  NN0
4 nnexpcl 10994 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
52, 3, 4mp2an 656 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
6 5nn 9759 . . . . . . . 8  |-  5  e.  NN
7 7nn 9761 . . . . . . . 8  |-  7  e.  NN
86, 7nnmulcli 9650 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
95, 8nnmulcli 9650 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
109nncni 9636 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
11 log2ublem1.3 . . . . . 6  |-  D  e. 
NN0
1211nn0cni 9856 . . . . 5  |-  D  e.  CC
13 log2ublem1.4 . . . . . 6  |-  E  e.  NN
1413nncni 9636 . . . . 5  |-  E  e.  CC
1513nnne0i 9660 . . . . 5  |-  E  =/=  0
1610, 12, 14, 15divassi 9396 . . . 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 9862 . . . . . . . . . 10  |-  3  e.  NN0
1918, 3nn0expcli 11007 . . . . . . . . 9  |-  ( 3 ^ 7 )  e. 
NN0
20 5nn0 9864 . . . . . . . . . 10  |-  5  e.  NN0
2120, 3nn0mulcli 9881 . . . . . . . . 9  |-  ( 5  x.  7 )  e. 
NN0
2219, 21nn0mulcli 9881 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e. 
NN0
2322, 11nn0mulcli 9881 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e. 
NN0
2423nn0rei 9855 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  e.  RR
25 log2ublem1.6 . . . . . . 7  |-  F  e. 
NN0
2625nn0rei 9855 . . . . . 6  |-  F  e.  RR
2713nnrei 9635 . . . . . . 7  |-  E  e.  RR
2813nngt0i 9659 . . . . . . 7  |-  0  <  E
2927, 28pm3.2i 443 . . . . . 6  |-  ( E  e.  RR  /\  0  <  E )
30 ledivmul 9509 . . . . . 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 3941 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  <_  F
349nnrei 9635 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
35 log2ublem1.2 . . . . 5  |-  A  e.  RR
3634, 35remulcli 8731 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  A )  e.  RR
3711nn0rei 9855 . . . . . 6  |-  D  e.  RR
38 nndivre 9661 . . . . . 6  |-  ( ( D  e.  RR  /\  E  e.  NN )  ->  ( D  /  E
)  e.  RR )
3937, 13, 38mp2an 656 . . . . 5  |-  ( D  /  E )  e.  RR
4034, 39remulcli 8731 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( D  /  E ) )  e.  RR
41 log2ublem1.5 . . . . 5  |-  B  e. 
NN0
4241nn0rei 9855 . . . 4  |-  B  e.  RR
4336, 40, 42, 26le2addi 9216 . . 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 5721 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  =  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( A  +  ( D  /  E ) ) )
4735recni 8729 . . . 4  |-  A  e.  CC
4839recni 8729 . . . 4  |-  ( D  /  E )  e.  CC
4910, 47, 48adddii 8727 . . 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 2274 . 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 3947 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 3920  (class class class)co 5710   RRcr 8616   0cc0 8617    + caddc 8620    x. cmul 8622    < clt 8747    <_ cle 8748    / cdiv 9303   NNcn 9626   3c3 9676   5c5 9678   7c7 9680   NN0cn0 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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