Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > log2ublem1 | Structured version Visualization version GIF version |
Description: Lemma for log2ub 25529. The proof of log2ub 25529, which is simply the evaluation of log2tlbnd 25525 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 11719 | . . . . . . . 8 ⊢ 3 ∈ ℕ | |
3 | 7nn0 11922 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 | |
4 | nnexpcl 13445 | . . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ) | |
5 | 2, 3, 4 | mp2an 690 | . . . . . . 7 ⊢ (3↑7) ∈ ℕ |
6 | 5nn 11726 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
7 | 7nn 11732 | . . . . . . . 8 ⊢ 7 ∈ ℕ | |
8 | 6, 7 | nnmulcli 11665 | . . . . . . 7 ⊢ (5 · 7) ∈ ℕ |
9 | 5, 8 | nnmulcli 11665 | . . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ |
10 | 9 | nncni 11650 | . . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ |
11 | log2ublem1.3 | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
12 | 11 | nn0cni 11912 | . . . . 5 ⊢ 𝐷 ∈ ℂ |
13 | log2ublem1.4 | . . . . . 6 ⊢ 𝐸 ∈ ℕ | |
14 | 13 | nncni 11650 | . . . . 5 ⊢ 𝐸 ∈ ℂ |
15 | 13 | nnne0i 11680 | . . . . 5 ⊢ 𝐸 ≠ 0 |
16 | 10, 12, 14, 15 | divassi 11398 | . . . 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 11918 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
19 | 18, 3 | nn0expcli 13458 | . . . . . . . . 9 ⊢ (3↑7) ∈ ℕ0 |
20 | 5nn0 11920 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
21 | 20, 3 | nn0mulcli 11938 | . . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ0 |
22 | 19, 21 | nn0mulcli 11938 | . . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ0 |
23 | 22, 11 | nn0mulcli 11938 | . . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0 |
24 | 23 | nn0rei 11911 | . . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ |
25 | log2ublem1.6 | . . . . . . 7 ⊢ 𝐹 ∈ ℕ0 | |
26 | 25 | nn0rei 11911 | . . . . . 6 ⊢ 𝐹 ∈ ℝ |
27 | 13 | nnrei 11649 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ |
28 | 13 | nngt0i 11679 | . . . . . . 7 ⊢ 0 < 𝐸 |
29 | 27, 28 | pm3.2i 473 | . . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸) |
30 | ledivmul 11518 | . . . . . 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 1457 | . . . . 5 ⊢ (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)) |
32 | 17, 31 | mpbir 233 | . . . 4 ⊢ ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 |
33 | 16, 32 | eqbrtrri 5091 | . . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹 |
34 | 9 | nnrei 11649 | . . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ |
35 | log2ublem1.2 | . . . . 5 ⊢ 𝐴 ∈ ℝ | |
36 | 34, 35 | remulcli 10659 | . . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ |
37 | 11 | nn0rei 11911 | . . . . . 6 ⊢ 𝐷 ∈ ℝ |
38 | nndivre 11681 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ) | |
39 | 37, 13, 38 | mp2an 690 | . . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ |
40 | 34, 39 | remulcli 10659 | . . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ |
41 | log2ublem1.5 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
42 | 41 | nn0rei 11911 | . . . 4 ⊢ 𝐵 ∈ ℝ |
43 | 36, 40, 42, 26 | le2addi 11205 | . . 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 690 | . 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 7169 | . . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) |
47 | 35 | recni 10657 | . . . 4 ⊢ 𝐴 ∈ ℂ |
48 | 39 | recni 10657 | . . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ |
49 | 10, 47, 48 | adddii 10655 | . . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) |
50 | 46, 49 | eqtr2i 2847 | . 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 5097 | 1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 / cdiv 11299 ℕcn 11640 3c3 11696 5c5 11698 7c7 11700 ℕ0cn0 11900 ↑cexp 13432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-exp 13433 |
This theorem is referenced by: log2ublem2 25527 log2ub 25529 |
Copyright terms: Public domain | W3C validator |