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Theorem log2ublem1 24473
Description: Lemma for log2ub 24476. The proof of log2ub 24476, which is simply the evaluation of log2tlbnd 24472 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 11063 . . . . . . . 8 3 ∈ ℕ
3 7nn0 11191 . . . . . . . 8 7 ∈ ℕ0
4 nnexpcl 12735 . . . . . . . 8 ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ)
52, 3, 4mp2an 704 . . . . . . 7 (3↑7) ∈ ℕ
6 5nn 11065 . . . . . . . 8 5 ∈ ℕ
7 7nn 11067 . . . . . . . 8 7 ∈ ℕ
86, 7nnmulcli 10921 . . . . . . 7 (5 · 7) ∈ ℕ
95, 8nnmulcli 10921 . . . . . 6 ((3↑7) · (5 · 7)) ∈ ℕ
109nncni 10907 . . . . 5 ((3↑7) · (5 · 7)) ∈ ℂ
11 log2ublem1.3 . . . . . 6 𝐷 ∈ ℕ0
1211nn0cni 11181 . . . . 5 𝐷 ∈ ℂ
13 log2ublem1.4 . . . . . 6 𝐸 ∈ ℕ
1413nncni 10907 . . . . 5 𝐸 ∈ ℂ
1513nnne0i 10932 . . . . 5 𝐸 ≠ 0
1610, 12, 14, 15divassi 10660 . . . 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 11187 . . . . . . . . . 10 3 ∈ ℕ0
1918, 3nn0expcli 12748 . . . . . . . . 9 (3↑7) ∈ ℕ0
20 5nn0 11189 . . . . . . . . . 10 5 ∈ ℕ0
2120, 3nn0mulcli 11208 . . . . . . . . 9 (5 · 7) ∈ ℕ0
2219, 21nn0mulcli 11208 . . . . . . . 8 ((3↑7) · (5 · 7)) ∈ ℕ0
2322, 11nn0mulcli 11208 . . . . . . 7 (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0
2423nn0rei 11180 . . . . . 6 (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ
25 log2ublem1.6 . . . . . . 7 𝐹 ∈ ℕ0
2625nn0rei 11180 . . . . . 6 𝐹 ∈ ℝ
2713nnrei 10906 . . . . . . 7 𝐸 ∈ ℝ
2813nngt0i 10931 . . . . . . 7 0 < 𝐸
2927, 28pm3.2i 470 . . . . . 6 (𝐸 ∈ ℝ ∧ 0 < 𝐸)
30 ledivmul 10778 . . . . . 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 1416 . . . . 5 (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹))
3217, 31mpbir 220 . . . 4 ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹
3316, 32eqbrtrri 4606 . . 3 (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹
349nnrei 10906 . . . . 5 ((3↑7) · (5 · 7)) ∈ ℝ
35 log2ublem1.2 . . . . 5 𝐴 ∈ ℝ
3634, 35remulcli 9933 . . . 4 (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ
3711nn0rei 11180 . . . . . 6 𝐷 ∈ ℝ
38 nndivre 10933 . . . . . 6 ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ)
3937, 13, 38mp2an 704 . . . . 5 (𝐷 / 𝐸) ∈ ℝ
4034, 39remulcli 9933 . . . 4 (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ
41 log2ublem1.5 . . . . 5 𝐵 ∈ ℕ0
4241nn0rei 11180 . . . 4 𝐵 ∈ ℝ
4336, 40, 42, 26le2addi 10470 . . 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 704 . 2 ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) ≤ (𝐵 + 𝐹)
45 log2ublem1.7 . . . 4 𝐶 = (𝐴 + (𝐷 / 𝐸))
4645oveq2i 6560 . . 3 (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸)))
4735recni 9931 . . . 4 𝐴 ∈ ℂ
4839recni 9931 . . . 4 (𝐷 / 𝐸) ∈ ℂ
4910, 47, 48adddii 9929 . . 3 (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)))
5046, 49eqtr2i 2633 . 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 4612 1 (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977   class class class wbr 4583  (class class class)co 6549  cr 9814  0cc0 9815   + caddc 9818   · cmul 9820   < clt 9953  cle 9954   / cdiv 10563  cn 10897  3c3 10948  5c5 10950  7c7 10952  0cn0 11169  cexp 12722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-exp 12723
This theorem is referenced by:  log2ublem2  24474  log2ub  24476
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