Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  log2ublem1 Structured version   Visualization version   GIF version

Theorem log2ublem1 25530
 Description: Lemma for log2ub 25533. The proof of log2ub 25533, which is simply the evaluation of log2tlbnd 25529 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) · 5 · 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) · (5 · 7)) · 𝐴) ≤ 𝐵
log2ublem1.2 𝐴 ∈ ℝ
log2ublem1.3 𝐷 ∈ ℕ0
log2ublem1.4 𝐸 ∈ ℕ
log2ublem1.5 𝐵 ∈ ℕ0
log2ublem1.6 𝐹 ∈ ℕ0
log2ublem1.7 𝐶 = (𝐴 + (𝐷 / 𝐸))
log2ublem1.8 (𝐵 + 𝐹) = 𝐺
log2ublem1.9 (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)
Assertion
Ref Expression
log2ublem1 (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺

Proof of Theorem log2ublem1
StepHypRef Expression
1 log2ublem1.1 . . 3 (((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵
2 3nn 11704 . . . . . . . 8 3 ∈ ℕ
3 7nn0 11907 . . . . . . . 8 7 ∈ ℕ0
4 nnexpcl 13438 . . . . . . . 8 ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ)
52, 3, 4mp2an 691 . . . . . . 7 (3↑7) ∈ ℕ
6 5nn 11711 . . . . . . . 8 5 ∈ ℕ
7 7nn 11717 . . . . . . . 8 7 ∈ ℕ
86, 7nnmulcli 11650 . . . . . . 7 (5 · 7) ∈ ℕ
95, 8nnmulcli 11650 . . . . . 6 ((3↑7) · (5 · 7)) ∈ ℕ
109nncni 11635 . . . . 5 ((3↑7) · (5 · 7)) ∈ ℂ
11 log2ublem1.3 . . . . . 6 𝐷 ∈ ℕ0
1211nn0cni 11897 . . . . 5 𝐷 ∈ ℂ
13 log2ublem1.4 . . . . . 6 𝐸 ∈ ℕ
1413nncni 11635 . . . . 5 𝐸 ∈ ℂ
1513nnne0i 11665 . . . . 5 𝐸 ≠ 0
1610, 12, 14, 15divassi 11385 . . . 4 ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) = (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))
17 log2ublem1.9 . . . . 5 (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)
18 3nn0 11903 . . . . . . . . . 10 3 ∈ ℕ0
1918, 3nn0expcli 13451 . . . . . . . . 9 (3↑7) ∈ ℕ0
20 5nn0 11905 . . . . . . . . . 10 5 ∈ ℕ0
2120, 3nn0mulcli 11923 . . . . . . . . 9 (5 · 7) ∈ ℕ0
2219, 21nn0mulcli 11923 . . . . . . . 8 ((3↑7) · (5 · 7)) ∈ ℕ0
2322, 11nn0mulcli 11923 . . . . . . 7 (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0
2423nn0rei 11896 . . . . . 6 (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25 log2ublem1.6 . . . . . . 7 𝐹 ∈ ℕ0
2625nn0rei 11896 . . . . . 6 𝐹 ∈ ℝ
2713nnrei 11634 . . . . . . 7 𝐸 ∈ ℝ
2813nngt0i 11664 . . . . . . 7 0 < 𝐸
2927, 28pm3.2i 474 . . . . . 6 (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30 ledivmul 11505 . . . . . 6 (((((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ ∧ 𝐹 ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)))
3124, 26, 29, 30mp3an 1458 . . . . 5 (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹))
3217, 31mpbir 234 . . . 4 ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹
3316, 32eqbrtrri 5065 . . 3 (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
349nnrei 11634 . . . . 5 ((3↑7) · (5 · 7)) ∈ ℝ
35 log2ublem1.2 . . . . 5 𝐴 ∈ ℝ
3634, 35remulcli 10646 . . . 4 (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
3711nn0rei 11896 . . . . . 6 𝐷 ∈ ℝ
38 nndivre 11666 . . . . . 6 ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
3937, 13, 38mp2an 691 . . . . 5 (𝐷 / 𝐸) ∈ ℝ
4034, 39remulcli 10646 . . . 4 (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41 log2ublem1.5 . . . . 5 𝐵 ∈ ℕ0
4241nn0rei 11896 . . . 4 𝐵 ∈ ℝ
4336, 40, 42, 26le2addi 11192 . . 3 (((((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵 ∧ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹) → ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) ≤ (𝐵 + 𝐹))
441, 33, 43mp2an 691 . 2 ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) ≤ (𝐵 + 𝐹)
45 log2ublem1.7 . . . 4 𝐶 = (𝐴 + (𝐷 / 𝐸))
4645oveq2i 7151 . . 3 (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
4735recni 10644 . . . 4 𝐴 ∈ ℂ
4839recni 10644 . . . 4 (𝐷 / 𝐸) ∈ ℂ
4910, 47, 48adddii 10642 . . 3 (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
5046, 49eqtr2i 2846 . 2 ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) = (((3↑7) · (5 · 7)) · 𝐶)
51 log2ublem1.8 . 2 (𝐵 + 𝐹) = 𝐺
5244, 50, 513brtr3i 5071 1 (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114   class class class wbr 5042  (class class class)co 7140  ℝcr 10525  0cc0 10526   + caddc 10529   · cmul 10531   < clt 10664   ≤ cle 10665   / cdiv 11286  ℕcn 11625  3c3 11681  5c5 11683  7c7 11685  ℕ0cn0 11885  ↑cexp 13425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-exp 13426 This theorem is referenced by:  log2ublem2  25531  log2ub  25533
 Copyright terms: Public domain W3C validator