Theorem log2ublem1 23776

Description: Lemma for log2ub 23779. The proof of log2ub 23779, which is simply the evaluation of log2tlbnd 23775 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 10757	. . . . . . . 8 |- 3 e. NN
3		7nn0 10880	. . . . . . . 8 |- 7 e. NN0
4		nnexpcl 12271	. . . . . . . 8 |- ( ( 3 e. NN /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. NN )
5	2, 3, 4	mp2an 676	. . . . . . 7 |- ( 3 ^ 7 ) e. NN
6		5nn 10759	. . . . . . . 8 |- 5 e. NN
7		7nn 10761	. . . . . . . 8 |- 7 e. NN
8	6, 7	nnmulcli 10622	. . . . . . 7 |- ( 5 x. 7 ) e. NN
9	5, 8	nnmulcli 10622	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN
10	9	nncni 10608	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
11		log2ublem1.3	. . . . . 6 |- D e. NN0
12	11	nn0cni 10870	. . . . 5 |- D e. CC
13		log2ublem1.4	. . . . . 6 |- E e. NN
14	13	nncni 10608	. . . . 5 |- E e. CC
15	13	nnne0i 10633	. . . . 5 |- E =/= 0
16	10, 12, 14, 15	divassi 10352	. . . 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 10876	. . . . . . . . . 10 |- 3 e. NN0
19	18, 3	nn0expcli 12284	. . . . . . . . 9 |- ( 3 ^ 7 ) e. NN0
20		5nn0 10878	. . . . . . . . . 10 |- 5 e. NN0
21	20, 3	nn0mulcli 10897	. . . . . . . . 9 |- ( 5 x. 7 ) e. NN0
22	19, 21	nn0mulcli 10897	. . . . . . . 8 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN0
23	22, 11	nn0mulcli 10897	. . . . . . 7 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. NN0
24	23	nn0rei 10869	. . . . . 6 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. RR
25		log2ublem1.6	. . . . . . 7 |- F e. NN0
26	25	nn0rei 10869	. . . . . 6 |- F e. RR
27	13	nnrei 10607	. . . . . . 7 |- E e. RR
28	13	nngt0i 10632	. . . . . . 7 |- 0 < E
29	27, 28	pm3.2i 456	. . . . . 6 |- ( E e. RR /\ 0 < E )
30		ledivmul 10470	. . . . . 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 1360	. . . . 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 212	. . . 4 |- ( ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) / E ) <_ F
33	16, 32	eqbrtrri 4438	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) <_ F
34	9	nnrei 10607	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. RR
35		log2ublem1.2	. . . . 5 |- A e. RR
36	34, 35	remulcli 9646	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. A ) e. RR
37	11	nn0rei 10869	. . . . . 6 |- D e. RR
38		nndivre 10634	. . . . . 6 |- ( ( D e. RR /\ E e. NN ) -> ( D / E ) e. RR )
39	37, 13, 38	mp2an 676	. . . . 5 |- ( D / E ) e. RR
40	34, 39	remulcli 9646	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) e. RR
41		log2ublem1.5	. . . . 5 |- B e. NN0
42	41	nn0rei 10869	. . . 4 |- B e. RR
43	36, 40, 42, 26	le2addi 10166	. . 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 676	. 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 6307	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) = ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( A + ( D / E ) ) )
47	35	recni 9644	. . . 4 |- A e. CC
48	39	recni 9644	. . . 4 |- ( D / E ) e. CC
49	10, 47, 48	adddii 9642	. . 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 2450	. 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 4444	1 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) <_ G

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1867   class class class wbr 4417  (class class class)co 6296   RRcr 9527   0cc0 9528   + caddc 9531   x. cmul 9533   < clt 9664   <_ cle 9665   / cdiv 10258   NNcn 10598   3c3 10649   5c5 10651   7c7 10653   NN0cn0 10858   ^cexp 12258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-n0 10859  df-z 10927  df-uz 11149  df-seq 12200  df-exp 12259

This theorem is referenced by:  log2ublem2  23777  log2ub  23779