Theorem log2ublem1 24673

Description: Lemma for log2ub 24676. The proof of log2ub 24676, which is simply the evaluation of log2tlbnd 24672 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 1 0) 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 11186	. . . . . . . 8 |- 3 e. NN
3		7nn0 11314	. . . . . . . 8 |- 7 e. NN0
4		nnexpcl 12873	. . . . . . . 8 |- ( ( 3 e. NN /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. NN )
5	2, 3, 4	mp2an 708	. . . . . . 7 |- ( 3 ^ 7 ) e. NN
6		5nn 11188	. . . . . . . 8 |- 5 e. NN
7		7nn 11190	. . . . . . . 8 |- 7 e. NN
8	6, 7	nnmulcli 11044	. . . . . . 7 |- ( 5 x. 7 ) e. NN
9	5, 8	nnmulcli 11044	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN
10	9	nncni 11030	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
11		log2ublem1.3	. . . . . 6 |- D e. NN0
12	11	nn0cni 11304	. . . . 5 |- D e. CC
13		log2ublem1.4	. . . . . 6 |- E e. NN
14	13	nncni 11030	. . . . 5 |- E e. CC
15	13	nnne0i 11055	. . . . 5 |- E =/= 0
16	10, 12, 14, 15	divassi 10781	. . . 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 11310	. . . . . . . . . 10 |- 3 e. NN0
19	18, 3	nn0expcli 12886	. . . . . . . . 9 |- ( 3 ^ 7 ) e. NN0
20		5nn0 11312	. . . . . . . . . 10 |- 5 e. NN0
21	20, 3	nn0mulcli 11331	. . . . . . . . 9 |- ( 5 x. 7 ) e. NN0
22	19, 21	nn0mulcli 11331	. . . . . . . 8 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN0
23	22, 11	nn0mulcli 11331	. . . . . . 7 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. NN0
24	23	nn0rei 11303	. . . . . 6 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. RR
25		log2ublem1.6	. . . . . . 7 |- F e. NN0
26	25	nn0rei 11303	. . . . . 6 |- F e. RR
27	13	nnrei 11029	. . . . . . 7 |- E e. RR
28	13	nngt0i 11054	. . . . . . 7 |- 0 < E
29	27, 28	pm3.2i 471	. . . . . 6 |- ( E e. RR /\ 0 < E )
30		ledivmul 10899	. . . . . 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 1424	. . . . 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 221	. . . 4 |- ( ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) / E ) <_ F
33	16, 32	eqbrtrri 4676	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) <_ F
34	9	nnrei 11029	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. RR
35		log2ublem1.2	. . . . 5 |- A e. RR
36	34, 35	remulcli 10054	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. A ) e. RR
37	11	nn0rei 11303	. . . . . 6 |- D e. RR
38		nndivre 11056	. . . . . 6 |- ( ( D e. RR /\ E e. NN ) -> ( D / E ) e. RR )
39	37, 13, 38	mp2an 708	. . . . 5 |- ( D / E ) e. RR
40	34, 39	remulcli 10054	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) e. RR
41		log2ublem1.5	. . . . 5 |- B e. NN0
42	41	nn0rei 11303	. . . 4 |- B e. RR
43	36, 40, 42, 26	le2addi 10591	. . 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 708	. 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 6661	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) = ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( A + ( D / E ) ) )
47	35	recni 10052	. . . 4 |- A e. CC
48	39	recni 10052	. . . 4 |- ( D / E ) e. CC
49	10, 47, 48	adddii 10050	. . 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 2645	. 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 4682	1 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) <_ G

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   3c3 11071   5c5 11073   7c7 11075   NN0cn0 11292   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  log2ublem2  24674  log2ub  24676