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| Mirrors > Home > MPE Home > Th. List > log2ublem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for log2ub 26992. The proof of log2ub 26992, which is simply the evaluation of log2tlbnd 26988 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 12345 | . . . . . . . 8 ⊢ 3 ∈ ℕ | |
| 3 | 7nn0 12548 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 | |
| 4 | nnexpcl 14115 | . . . . . . . 8 ⊢ ((3 ∈ ℕ ∧ 7 ∈ ℕ0) → (3↑7) ∈ ℕ) | |
| 5 | 2, 3, 4 | mp2an 692 | . . . . . . 7 ⊢ (3↑7) ∈ ℕ |
| 6 | 5nn 12352 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
| 7 | 7nn 12358 | . . . . . . . 8 ⊢ 7 ∈ ℕ | |
| 8 | 6, 7 | nnmulcli 12291 | . . . . . . 7 ⊢ (5 · 7) ∈ ℕ |
| 9 | 5, 8 | nnmulcli 12291 | . . . . . 6 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ |
| 10 | 9 | nncni 12276 | . . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℂ |
| 11 | log2ublem1.3 | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
| 12 | 11 | nn0cni 12538 | . . . . 5 ⊢ 𝐷 ∈ ℂ |
| 13 | log2ublem1.4 | . . . . . 6 ⊢ 𝐸 ∈ ℕ | |
| 14 | 13 | nncni 12276 | . . . . 5 ⊢ 𝐸 ∈ ℂ |
| 15 | 13 | nnne0i 12306 | . . . . 5 ⊢ 𝐸 ≠ 0 |
| 16 | 10, 12, 14, 15 | divassi 12023 | . . . 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 12544 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ0 | |
| 19 | 18, 3 | nn0expcli 14129 | . . . . . . . . 9 ⊢ (3↑7) ∈ ℕ0 |
| 20 | 5nn0 12546 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
| 21 | 20, 3 | nn0mulcli 12564 | . . . . . . . . 9 ⊢ (5 · 7) ∈ ℕ0 |
| 22 | 19, 21 | nn0mulcli 12564 | . . . . . . . 8 ⊢ ((3↑7) · (5 · 7)) ∈ ℕ0 |
| 23 | 22, 11 | nn0mulcli 12564 | . . . . . . 7 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℕ0 |
| 24 | 23 | nn0rei 12537 | . . . . . 6 ⊢ (((3↑7) · (5 · 7)) · 𝐷) ∈ ℝ |
| 25 | log2ublem1.6 | . . . . . . 7 ⊢ 𝐹 ∈ ℕ0 | |
| 26 | 25 | nn0rei 12537 | . . . . . 6 ⊢ 𝐹 ∈ ℝ |
| 27 | 13 | nnrei 12275 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ |
| 28 | 13 | nngt0i 12305 | . . . . . . 7 ⊢ 0 < 𝐸 |
| 29 | 27, 28 | pm3.2i 470 | . . . . . 6 ⊢ (𝐸 ∈ ℝ ∧ 0 < 𝐸) |
| 30 | ledivmul 12144 | . . . . . 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 1463 | . . . . 5 ⊢ (((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 ↔ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)) |
| 32 | 17, 31 | mpbir 231 | . . . 4 ⊢ ((((3↑7) · (5 · 7)) · 𝐷) / 𝐸) ≤ 𝐹 |
| 33 | 16, 32 | eqbrtrri 5166 | . . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ≤ 𝐹 |
| 34 | 9 | nnrei 12275 | . . . . 5 ⊢ ((3↑7) · (5 · 7)) ∈ ℝ |
| 35 | log2ublem1.2 | . . . . 5 ⊢ 𝐴 ∈ ℝ | |
| 36 | 34, 35 | remulcli 11277 | . . . 4 ⊢ (((3↑7) · (5 · 7)) · 𝐴) ∈ ℝ |
| 37 | 11 | nn0rei 12537 | . . . . . 6 ⊢ 𝐷 ∈ ℝ |
| 38 | nndivre 12307 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℕ) → (𝐷 / 𝐸) ∈ ℝ) | |
| 39 | 37, 13, 38 | mp2an 692 | . . . . 5 ⊢ (𝐷 / 𝐸) ∈ ℝ |
| 40 | 34, 39 | remulcli 11277 | . . . 4 ⊢ (((3↑7) · (5 · 7)) · (𝐷 / 𝐸)) ∈ ℝ |
| 41 | log2ublem1.5 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
| 42 | 41 | nn0rei 12537 | . . . 4 ⊢ 𝐵 ∈ ℝ |
| 43 | 36, 40, 42, 26 | le2addi 11826 | . . 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 692 | . 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 7442 | . . 3 ⊢ (((3↑7) · (5 · 7)) · 𝐶) = (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) |
| 47 | 35 | recni 11275 | . . . 4 ⊢ 𝐴 ∈ ℂ |
| 48 | 39 | recni 11275 | . . . 4 ⊢ (𝐷 / 𝐸) ∈ ℂ |
| 49 | 10, 47, 48 | adddii 11273 | . . 3 ⊢ (((3↑7) · (5 · 7)) · (𝐴 + (𝐷 / 𝐸))) = ((((3↑7) · (5 · 7)) · 𝐴) + (((3↑7) · (5 · 7)) · (𝐷 / 𝐸))) |
| 50 | 46, 49 | eqtr2i 2766 | . 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 5172 | 1 ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 / cdiv 11920 ℕcn 12266 3c3 12322 5c5 12324 7c7 12326 ℕ0cn0 12526 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: log2ublem2 26990 log2ub 26992 |
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