Nearby theorems
|
|
Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > 3lexlogpow5ineq4 | Structured version Visualization version GIF version | ||
| Description: Sharper logarithm inequality chain. (Contributed by metakunt, 21-Aug-2024.) |
| Ref | Expression |
|---|---|
| 3lexlogpow5ineq4.1 | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
| 3lexlogpow5ineq4.2 | ⊢ (𝜑 → 3 ≤ 𝑋) |
| Ref | Expression |
|---|---|
| 3lexlogpow5ineq4 | ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 9re 12285 | . . 3 ⊢ 9 ∈ ℝ | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 9 ∈ ℝ) |
| 3 | 1nn0 12458 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 4 | 1nn 12197 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
| 5 | 3, 4 | decnncl 12669 | . . . . . 6 ⊢ ;11 ∈ ℕ |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → ;11 ∈ ℕ) |
| 7 | 6 | nnred 12201 | . . . 4 ⊢ (𝜑 → ;11 ∈ ℝ) |
| 8 | 7re 12279 | . . . . 5 ⊢ 7 ∈ ℝ | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 7 ∈ ℝ) |
| 10 | 0red 11177 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 11 | 7pos 12297 | . . . . . . 7 ⊢ 0 < 7 | |
| 12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 7) |
| 13 | 10, 12 | ltned 11310 | . . . . 5 ⊢ (𝜑 → 0 ≠ 7) |
| 14 | 13 | necomd 2980 | . . . 4 ⊢ (𝜑 → 7 ≠ 0) |
| 15 | 7, 9, 14 | redivcld 12010 | . . 3 ⊢ (𝜑 → (;11 / 7) ∈ ℝ) |
| 16 | 5nn0 12462 | . . . 4 ⊢ 5 ∈ ℕ0 | |
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 5 ∈ ℕ0) |
| 18 | 15, 17 | reexpcld 14128 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ∈ ℝ) |
| 19 | 2re 12260 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
| 21 | 2pos 12289 | . . . . 5 ⊢ 0 < 2 | |
| 22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 < 2) |
| 23 | 3lexlogpow5ineq4.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 24 | 3re 12266 | . . . . . 6 ⊢ 3 ∈ ℝ | |
| 25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → 3 ∈ ℝ) |
| 26 | 3pos 12291 | . . . . . 6 ⊢ 0 < 3 | |
| 27 | 26 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 < 3) |
| 28 | 3lexlogpow5ineq4.2 | . . . . 5 ⊢ (𝜑 → 3 ≤ 𝑋) | |
| 29 | 10, 25, 23, 27, 28 | ltletrd 11334 | . . . 4 ⊢ (𝜑 → 0 < 𝑋) |
| 30 | 1red 11175 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 31 | 1lt2 12352 | . . . . . . 7 ⊢ 1 < 2 | |
| 32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 < 2) |
| 33 | 30, 32 | ltned 11310 | . . . . 5 ⊢ (𝜑 → 1 ≠ 2) |
| 34 | 33 | necomd 2980 | . . . 4 ⊢ (𝜑 → 2 ≠ 1) |
| 35 | 20, 22, 23, 29, 34 | relogbcld 41961 | . . 3 ⊢ (𝜑 → (2 logb 𝑋) ∈ ℝ) |
| 36 | 35, 17 | reexpcld 14128 | . 2 ⊢ (𝜑 → ((2 logb 𝑋)↑5) ∈ ℝ) |
| 37 | 3lexlogpow5ineq1 42042 | . . 3 ⊢ 9 < ((;11 / 7)↑5) | |
| 38 | 37 | a1i 11 | . 2 ⊢ (𝜑 → 9 < ((;11 / 7)↑5)) |
| 39 | 23, 28 | 3lexlogpow5ineq2 42043 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) |
| 40 | 2, 18, 36, 38, 39 | ltletrd 11334 | 1 ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2109 class class class wbr 5107 (class class class)co 7387 ℝcr 11067 0cc0 11068 1c1 11069 < clt 11208 ≤ cle 11209 / cdiv 11835 ℕcn 12186 2c2 12241 3c3 12242 5c5 12244 7c7 12246 9c9 12248 ℕ0cn0 12442 ;cdc 12649 ↑cexp 14026 logb clogb 26674 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-cxp 26466 df-logb 26675 |
| This theorem is referenced by: 3lexlogpow5ineq3 42045 aks4d1lem1 42050 aks4d1p1 42064 aks4d1p6 42069 aks4d1p7d1 42070 aks4d1p7 42071 aks4d1p8 42075 |
| Copyright terms: Public domain | W3C validator |