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Theorem log2ublem1 26894
Description: Lemma for log2ub 26897. The proof of log2ub 26897, which is simply the evaluation of log2tlbnd 26893 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 12319 . . . . . . . 8 3 ∈ ℕ
3 7nn0 12522 . . . . . . . 8 7 ∈ ℕ0
4 nnexpcl 14069 . . . . . . . 8 ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ)
52, 3, 4mp2an 690 . . . . . . 7 (3↑7) ∈ ℕ
6 5nn 12326 . . . . . . . 8 5 ∈ ℕ
7 7nn 12332 . . . . . . . 8 7 ∈ ℕ
86, 7nnmulcli 12265 . . . . . . 7 (5 · 7) ∈ ℕ
95, 8nnmulcli 12265 . . . . . 6 ((3↑7) · (5 · 7)) ∈ ℕ
109nncni 12250 . . . . 5 ((3↑7) · (5 · 7)) ∈ ℂ
11 log2ublem1.3 . . . . . 6 𝐷 ∈ ℕ0
1211nn0cni 12512 . . . . 5 𝐷 ∈ ℂ
13 log2ublem1.4 . . . . . 6 𝐸 ∈ ℕ
1413nncni 12250 . . . . 5 𝐸 ∈ ℂ
1513nnne0i 12280 . . . . 5 𝐸 ≠ 0
1610, 12, 14, 15divassi 11998 . . . 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 12518 . . . . . . . . . 10 3 ∈ ℕ0
1918, 3nn0expcli 14083 . . . . . . . . 9 (3↑7) ∈ ℕ0
20 5nn0 12520 . . . . . . . . . 10 5 ∈ ℕ0
2120, 3nn0mulcli 12538 . . . . . . . . 9 (5 · 7) ∈ ℕ0
2219, 21nn0mulcli 12538 . . . . . . . 8 ((3↑7) · (5 · 7)) ∈ ℕ0
2322, 11nn0mulcli 12538 . . . . . . 7 (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0
2423nn0rei 12511 . . . . . 6 (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25 log2ublem1.6 . . . . . . 7 𝐹 ∈ ℕ0
2625nn0rei 12511 . . . . . 6 𝐹 ∈ ℝ
2713nnrei 12249 . . . . . . 7 𝐸 ∈ ℝ
2813nngt0i 12279 . . . . . . 7 0 < 𝐸
2927, 28pm3.2i 469 . . . . . 6 (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30 ledivmul 12118 . . . . . 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 1457 . . . . 5 (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹))
3217, 31mpbir 230 . . . 4 ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹
3316, 32eqbrtrri 5166 . . 3 (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
349nnrei 12249 . . . . 5 ((3↑7) · (5 · 7)) ∈ ℝ
35 log2ublem1.2 . . . . 5 𝐴 ∈ ℝ
3634, 35remulcli 11258 . . . 4 (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
3711nn0rei 12511 . . . . . 6 𝐷 ∈ ℝ
38 nndivre 12281 . . . . . 6 ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
3937, 13, 38mp2an 690 . . . . 5 (𝐷 / 𝐸) ∈ ℝ
4034, 39remulcli 11258 . . . 4 (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41 log2ublem1.5 . . . . 5 𝐵 ∈ ℕ0
4241nn0rei 12511 . . . 4 𝐵 ∈ ℝ
4336, 40, 42, 26le2addi 11805 . . 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 690 . 2 ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) ≤ (𝐵 + 𝐹)
45 log2ublem1.7 . . . 4 𝐶 = (𝐴 + (𝐷 / 𝐸))
4645oveq2i 7426 . . 3 (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
4735recni 11256 . . . 4 𝐴 ∈ ℂ
4839recni 11256 . . . 4 (𝐷 / 𝐸) ∈ ℂ
4910, 47, 48adddii 11254 . . 3 (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
5046, 49eqtr2i 2754 . 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 5172 1 (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098   class class class wbr 5143  (class class class)co 7415  cr 11135  0cc0 11136   + caddc 11139   · cmul 11141   < clt 11276  cle 11277   / cdiv 11899  cn 12240  3c3 12296  5c5 12298  7c7 12300  0cn0 12500  cexp 14056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-n0 12501  df-z 12587  df-uz 12851  df-seq 13997  df-exp 14057
This theorem is referenced by:  log2ublem2  26895  log2ub  26897
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