Theorem log2ublem1 22284

Description: Lemma for log2ub 22287. The proof of log2ub 22287, which is simply the evaluation of log2tlbnd 22283 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 10476	. . . . . . . 8 |- 3 e. NN
3		7nn0 10597	. . . . . . . 8 |- 7 e. NN0
4		nnexpcl 11874	. . . . . . . 8 |- ( ( 3 e. NN /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. NN )
5	2, 3, 4	mp2an 667	. . . . . . 7 |- ( 3 ^ 7 ) e. NN
6		5nn 10478	. . . . . . . 8 |- 5 e. NN
7		7nn 10480	. . . . . . . 8 |- 7 e. NN
8	6, 7	nnmulcli 10342	. . . . . . 7 |- ( 5 x. 7 ) e. NN
9	5, 8	nnmulcli 10342	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN
10	9	nncni 10328	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
11		log2ublem1.3	. . . . . 6 |- D e. NN0
12	11	nn0cni 10587	. . . . 5 |- D e. CC
13		log2ublem1.4	. . . . . 6 |- E e. NN
14	13	nncni 10328	. . . . 5 |- E e. CC
15	13	nnne0i 10352	. . . . 5 |- E =/= 0
16	10, 12, 14, 15	divassi 10083	. . . 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 10593	. . . . . . . . . 10 |- 3 e. NN0
19	18, 3	nn0expcli 11887	. . . . . . . . 9 |- ( 3 ^ 7 ) e. NN0
20		5nn0 10595	. . . . . . . . . 10 |- 5 e. NN0
21	20, 3	nn0mulcli 10614	. . . . . . . . 9 |- ( 5 x. 7 ) e. NN0
22	19, 21	nn0mulcli 10614	. . . . . . . 8 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN0
23	22, 11	nn0mulcli 10614	. . . . . . 7 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. NN0
24	23	nn0rei 10586	. . . . . 6 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. RR
25		log2ublem1.6	. . . . . . 7 |- F e. NN0
26	25	nn0rei 10586	. . . . . 6 |- F e. RR
27	13	nnrei 10327	. . . . . . 7 |- E e. RR
28	13	nngt0i 10351	. . . . . . 7 |- 0 < E
29	27, 28	pm3.2i 452	. . . . . 6 |- ( E e. RR /\ 0 < E )
30		ledivmul 10201	. . . . . 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 1309	. . . . 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 209	. . . 4 |- ( ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) / E ) <_ F
33	16, 32	eqbrtrri 4310	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) <_ F
34	9	nnrei 10327	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. RR
35		log2ublem1.2	. . . . 5 |- A e. RR
36	34, 35	remulcli 9396	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. A ) e. RR
37	11	nn0rei 10586	. . . . . 6 |- D e. RR
38		nndivre 10353	. . . . . 6 |- ( ( D e. RR /\ E e. NN ) -> ( D / E ) e. RR )
39	37, 13, 38	mp2an 667	. . . . 5 |- ( D / E ) e. RR
40	34, 39	remulcli 9396	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) e. RR
41		log2ublem1.5	. . . . 5 |- B e. NN0
42	41	nn0rei 10586	. . . 4 |- B e. RR
43	36, 40, 42, 26	le2addi 9899	. . 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 667	. 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 6101	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) = ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( A + ( D / E ) ) )
47	35	recni 9394	. . . 4 |- A e. CC
48	39	recni 9394	. . . 4 |- ( D / E ) e. CC
49	10, 47, 48	adddii 9392	. . 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 2462	. 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 4316	1 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) <_ G

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   class class class wbr 4289  (class class class)co 6090   RRcr 9277   0cc0 9278   + caddc 9281   x. cmul 9283   < clt 9414   <_ cle 9415   / cdiv 9989   NNcn 10318   3c3 10368   5c5 10370   7c7 10372   NN0cn0 10575   ^cexp 11861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-n0 10576  df-z 10643  df-uz 10858  df-seq 11803  df-exp 11862

This theorem is referenced by:  log2ublem2  22285  log2ub  22287