Theorem log2ublem1 23928

Description: Lemma for log2ub 23931. The proof of log2ub 23931, which is simply the evaluation of log2tlbnd 23927 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

Step	Hyp	Ref	Expression
1		log2ublem1.1	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. A ) <_ B
2		3nn 10802	. . . . . . . 8 |- 3 e. NN
3		7nn0 10925	. . . . . . . 8 |- 7 e. NN0
4		nnexpcl 12323	. . . . . . . 8 |- ( ( 3 e. NN /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. NN )
5	2, 3, 4	mp2an 683	. . . . . . 7 |- ( 3 ^ 7 ) e. NN
6		5nn 10804	. . . . . . . 8 |- 5 e. NN
7		7nn 10806	. . . . . . . 8 |- 7 e. NN
8	6, 7	nnmulcli 10666	. . . . . . 7 |- ( 5 x. 7 ) e. NN
9	5, 8	nnmulcli 10666	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN
10	9	nncni 10652	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
11		log2ublem1.3	. . . . . 6 |- D e. NN0
12	11	nn0cni 10915	. . . . 5 |- D e. CC
13		log2ublem1.4	. . . . . 6 |- E e. NN
14	13	nncni 10652	. . . . 5 |- E e. CC
15	13	nnne0i 10677	. . . . 5 |- E =/= 0
16	10, 12, 14, 15	divassi 10396	. . . 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 10921	. . . . . . . . . 10 |- 3 e. NN0
19	18, 3	nn0expcli 12336	. . . . . . . . 9 |- ( 3 ^ 7 ) e. NN0
20		5nn0 10923	. . . . . . . . . 10 |- 5 e. NN0
21	20, 3	nn0mulcli 10942	. . . . . . . . 9 |- ( 5 x. 7 ) e. NN0
22	19, 21	nn0mulcli 10942	. . . . . . . 8 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN0
23	22, 11	nn0mulcli 10942	. . . . . . 7 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. NN0
24	23	nn0rei 10914	. . . . . 6 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. RR
25		log2ublem1.6	. . . . . . 7 |- F e. NN0
26	25	nn0rei 10914	. . . . . 6 |- F e. RR
27	13	nnrei 10651	. . . . . . 7 |- E e. RR
28	13	nngt0i 10676	. . . . . . 7 |- 0 < E
29	27, 28	pm3.2i 461	. . . . . 6 |- ( E e. RR /\ 0 < E )
30		ledivmul 10514	. . . . . 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 ) ) )
31	24, 26, 29, 30	mp3an 1373	. . . . 5 |- ( ( ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) / E ) <_ F <-> ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) <_ ( E x. F ) )
32	17, 31	mpbir 214	. . . 4 |- ( ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) / E ) <_ F
33	16, 32	eqbrtrri 4440	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) <_ F
34	9	nnrei 10651	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. RR
35		log2ublem1.2	. . . . 5 |- A e. RR
36	34, 35	remulcli 9688	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. A ) e. RR
37	11	nn0rei 10914	. . . . . 6 |- D e. RR
38		nndivre 10678	. . . . . 6 |- ( ( D e. RR /\ E e. NN ) -> ( D / E ) e. RR )
39	37, 13, 38	mp2an 683	. . . . 5 |- ( D / E ) e. RR
40	34, 39	remulcli 9688	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) e. RR
41		log2ublem1.5	. . . . 5 |- B e. NN0
42	41	nn0rei 10914	. . . 4 |- B e. RR
43	36, 40, 42, 26	le2addi 10210	. . 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 ) )
44	1, 33, 43	mp2an 683	. 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 ) )
46	45	oveq2i 6331	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) = ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( A + ( D / E ) ) )
47	35	recni 9686	. . . 4 |- A e. CC
48	39	recni 9686	. . . 4 |- ( D / E ) e. CC
49	10, 47, 48	adddii 9684	. . 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 ) ) )
50	46, 49	eqtr2i 2485	. 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
52	44, 50, 51	3brtr3i 4446	1 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) <_ G

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   class class class wbr 4418  (class class class)co 6320   RRcr 9569   0cc0 9570   + caddc 9573   x. cmul 9575   < clt 9706   <_ cle 9707   / cdiv 10302   NNcn 10642   3c3 10693   5c5 10695   7c7 10697   NN0cn0 10903   ^cexp 12310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-7 10706  df-n0 10904  df-z 10972  df-uz 11194  df-seq 12252  df-exp 12311

This theorem is referenced by:  log2ublem2  23929  log2ub  23931