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Mirrors > Home > MPE Home > Th. List > log2ublem1 | Structured version Visualization version GIF version |
Description: Lemma for log2ub 25132. The proof of log2ub 25132, which is simply the evaluation of log2tlbnd 25128 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.) |
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) |
Ref | Expression |
---|---|
log2ublem1 | ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | log2ublem1.1 | . . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵 | |
2 | 3nn 11458 | . . . . . . . 8 ⊢ 3 ∈ ℕ | |
3 | 7nn0 11670 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 | |
4 | nnexpcl 13195 | . . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ) | |
5 | 2, 3, 4 | mp2an 682 | . . . . . . 7 ⊢ (3↑7) ∈ ℕ |
6 | 5nn 11467 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
7 | 7nn 11475 | . . . . . . . 8 ⊢ 7 ∈ ℕ | |
8 | 6, 7 | nnmulcli 11405 | . . . . . . 7 ⊢ (5 · 7) ∈ ℕ |
9 | 5, 8 | nnmulcli 11405 | . . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ |
10 | 9 | nncni 11389 | . . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ |
11 | log2ublem1.3 | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
12 | 11 | nn0cni 11659 | . . . . 5 ⊢ 𝐷 ∈ ℂ |
13 | log2ublem1.4 | . . . . . 6 ⊢ 𝐸 ∈ ℕ | |
14 | 13 | nncni 11389 | . . . . 5 ⊢ 𝐸 ∈ ℂ |
15 | 13 | nnne0i 11419 | . . . . 5 ⊢ 𝐸 ≠ 0 |
16 | 10, 12, 14, 15 | divassi 11133 | . . . 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 11666 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
19 | 18, 3 | nn0expcli 13208 | . . . . . . . . 9 ⊢ (3↑7) ∈ ℕ0 |
20 | 5nn0 11668 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
21 | 20, 3 | nn0mulcli 11686 | . . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ0 |
22 | 19, 21 | nn0mulcli 11686 | . . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ0 |
23 | 22, 11 | nn0mulcli 11686 | . . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0 |
24 | 23 | nn0rei 11658 | . . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ |
25 | log2ublem1.6 | . . . . . . 7 ⊢ 𝐹 ∈ ℕ0 | |
26 | 25 | nn0rei 11658 | . . . . . 6 ⊢ 𝐹 ∈ ℝ |
27 | 13 | nnrei 11388 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ |
28 | 13 | nngt0i 11418 | . . . . . . 7 ⊢ 0 < 𝐸 |
29 | 27, 28 | pm3.2i 464 | . . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸) |
30 | ledivmul 11255 | . . . . . 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))) | |
31 | 24, 26, 29, 30 | mp3an 1534 | . . . . 5 ⊢ (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)) |
32 | 17, 31 | mpbir 223 | . . . 4 ⊢ ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 |
33 | 16, 32 | eqbrtrri 4911 | . . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹 |
34 | 9 | nnrei 11388 | . . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ |
35 | log2ublem1.2 | . . . . 5 ⊢ 𝐴 ∈ ℝ | |
36 | 34, 35 | remulcli 10395 | . . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ |
37 | 11 | nn0rei 11658 | . . . . . 6 ⊢ 𝐷 ∈ ℝ |
38 | nndivre 11420 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ) | |
39 | 37, 13, 38 | mp2an 682 | . . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ |
40 | 34, 39 | remulcli 10395 | . . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ |
41 | log2ublem1.5 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
42 | 41 | nn0rei 11658 | . . . 4 ⊢ 𝐵 ∈ ℝ |
43 | 36, 40, 42, 26 | le2addi 10940 | . . 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)) |
44 | 1, 33, 43 | mp2an 682 | . 2 ⊢ ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) ≤ (𝐵 + 𝐹) |
45 | log2ublem1.7 | . . . 4 ⊢ 𝐶 = (𝐴 + (𝐷 / 𝐸)) | |
46 | 45 | oveq2i 6935 | . . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) |
47 | 35 | recni 10393 | . . . 4 ⊢ 𝐴 ∈ ℂ |
48 | 39 | recni 10393 | . . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ |
49 | 10, 47, 48 | adddii 10391 | . . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) |
50 | 46, 49 | eqtr2i 2803 | . 2 ⊢ ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) = (((3↑7) · (5 · 7)) · 𝐶) |
51 | log2ublem1.8 | . 2 ⊢ (𝐵 + 𝐹) = 𝐺 | |
52 | 44, 50, 51 | 3brtr3i 4917 | 1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 (class class class)co 6924 ℝcr 10273 0cc0 10274 + caddc 10277 · cmul 10279 < clt 10413 ≤ cle 10414 / cdiv 11034 ℕcn 11378 3c3 11435 5c5 11437 7c7 11439 ℕ0cn0 11646 ↑cexp 13182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-n0 11647 df-z 11733 df-uz 11997 df-seq 13124 df-exp 13183 |
This theorem is referenced by: log2ublem2 25130 log2ub 25132 |
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