Theorem log2ublem1 16083

Description: Lemma for log2ublog2 16086. The proof of log2ublog2 16086, which is simply the evaluation of log2tlbndlog2 16082 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 9467	. . . . . . . 8 |- 3 e. NN
3		7nn0 9585	. . . . . . . 8 |- 7 e. NN0
4		nnexpcl 10989	. . . . . . . 8 |- ( ( 3 e. NN /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. NN )
5	2, 3, 4	mp2an 430	. . . . . . 7 |- ( 3 ^ 7 ) e. NN
6		5nn 9469	. . . . . . . 8 |- 5 e. NN
7		7nn 9471	. . . . . . . 8 |- 7 e. NN
8	6, 7	nnmulcli 9326	. . . . . . 7 |- ( 5 x. 7 ) e. NN
9	5, 8	nnmulcli 9326	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN
10	9	nncni 9314	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
11		log2ublem1.3	. . . . . 6 |- D e. NN0
12	11	nn0cni 9575	. . . . 5 |- D e. CC
13		log2ublem1.4	. . . . . 6 |- E e. NN
14	13	nncni 9314	. . . . 5 |- E e. CC
15	13	nnap0i 9335	. . . . 5 |- E # 0
16	10, 12, 14, 15	divassapi 9098	. . . 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 9581	. . . . . . . . . 10 |- 3 e. NN0
19	18, 3	nn0expcli 11002	. . . . . . . . 9 |- ( 3 ^ 7 ) e. NN0
20		5nn0 9583	. . . . . . . . . 10 |- 5 e. NN0
21	20, 3	nn0mulcli 9601	. . . . . . . . 9 |- ( 5 x. 7 ) e. NN0
22	19, 21	nn0mulcli 9601	. . . . . . . 8 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN0
23	22, 11	nn0mulcli 9601	. . . . . . 7 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. NN0
24	23	nn0rei 9574	. . . . . 6 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) e. RR
25		log2ublem1.6	. . . . . . 7 |- F e. NN0
26	25	nn0rei 9574	. . . . . 6 |- F e. RR
27	13	nnrei 9313	. . . . . . 7 |- E e. RR
28	13	nngt0i 9334	. . . . . . 7 |- 0 < E
29	27, 28	pm3.2i 272	. . . . . 6 |- ( E e. RR /\ 0 < E )
30		ledivmul 9207	. . . . . 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 1378	. . . . 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 146	. . . 4 |- ( ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. D ) / E ) <_ F
33	16, 32	eqbrtrri 4153	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) <_ F
34	9	nnrei 9313	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. RR
35		log2ublem1.2	. . . . 5 |- A e. RR
36	34, 35	remulcli 8340	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. A ) e. RR
37	11	nn0rei 9574	. . . . . 6 |- D e. RR
38		nndivre 9340	. . . . . 6 |- ( ( D e. RR /\ E e. NN ) -> ( D / E ) e. RR )
39	37, 13, 38	mp2an 430	. . . . 5 |- ( D / E ) e. RR
40	34, 39	remulcli 8340	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( D / E ) ) e. RR
41		log2ublem1.5	. . . . 5 |- B e. NN0
42	41	nn0rei 9574	. . . 4 |- B e. RR
43	36, 40, 42, 26	le2addi 8839	. . 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 430	. 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 6096	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) = ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. ( A + ( D / E ) ) )
47	35	recni 8338	. . . 4 |- A e. CC
48	39	recni 8338	. . . 4 |- ( D / E ) e. CC
49	10, 47, 48	adddii 8336	. . 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 2260	. 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 4159	1 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. C ) <_ G

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   / cdiv 9002   NNcn 9304   3c3 9356   5c5 9358   7c7 9360   NN0cn0 9563   ^cexp 10975

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-exp 10976

This theorem is used by:  log2ublem2  16084  log2ublog2  16086