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 12258 | . . 3 ⊢ 9 ∈ ℝ | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 9 ∈ ℝ) |
| 3 | 1nn0 12431 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 4 | 1nn 12170 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
| 5 | 3, 4 | decnncl 12641 | . . . . . 6 ⊢ ;11 ∈ ℕ |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → ;11 ∈ ℕ) |
| 7 | 6 | nnred 12174 | . . . 4 ⊢ (𝜑 → ;11 ∈ ℝ) |
| 8 | 7re 12252 | . . . . 5 ⊢ 7 ∈ ℝ | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 7 ∈ ℝ) |
| 10 | 0red 11149 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 11 | 7pos 12270 | . . . . . . 7 ⊢ 0 < 7 | |
| 12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 7) |
| 13 | 10, 12 | ltned 11283 | . . . . 5 ⊢ (𝜑 → 0 ≠ 7) |
| 14 | 13 | necomd 2988 | . . . 4 ⊢ (𝜑 → 7 ≠ 0) |
| 15 | 7, 9, 14 | redivcld 11983 | . . 3 ⊢ (𝜑 → (;11 / 7) ∈ ℝ) |
| 16 | 5nn0 12435 | . . . 4 ⊢ 5 ∈ ℕ0 | |
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 5 ∈ ℕ0) |
| 18 | 15, 17 | reexpcld 14100 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ∈ ℝ) |
| 19 | 2re 12233 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
| 21 | 2pos 12262 | . . . . 5 ⊢ 0 < 2 | |
| 22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 < 2) |
| 23 | 3lexlogpow5ineq4.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 24 | 3re 12239 | . . . . . 6 ⊢ 3 ∈ ℝ | |
| 25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → 3 ∈ ℝ) |
| 26 | 3pos 12264 | . . . . . 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 11307 | . . . 4 ⊢ (𝜑 → 0 < 𝑋) |
| 30 | 1red 11147 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 31 | 1lt2 12325 | . . . . . . 7 ⊢ 1 < 2 | |
| 32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 < 2) |
| 33 | 30, 32 | ltned 11283 | . . . . 5 ⊢ (𝜑 → 1 ≠ 2) |
| 34 | 33 | necomd 2988 | . . . 4 ⊢ (𝜑 → 2 ≠ 1) |
| 35 | 20, 22, 23, 29, 34 | relogbcld 42372 | . . 3 ⊢ (𝜑 → (2 logb 𝑋) ∈ ℝ) |
| 36 | 35, 17 | reexpcld 14100 | . 2 ⊢ (𝜑 → ((2 logb 𝑋)↑5) ∈ ℝ) |
| 37 | 3lexlogpow5ineq1 42453 | . . 3 ⊢ 9 < ((;11 / 7)↑5) | |
| 38 | 37 | a1i 11 | . 2 ⊢ (𝜑 → 9 < ((;11 / 7)↑5)) |
| 39 | 23, 28 | 3lexlogpow5ineq2 42454 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) |
| 40 | 2, 18, 36, 38, 39 | ltletrd 11307 | 1 ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5100 (class class class)co 7370 ℝcr 11039 0cc0 11040 1c1 11041 < clt 11180 ≤ cle 11181 / cdiv 11808 ℕcn 12159 2c2 12214 3c3 12215 5c5 12217 7c7 12219 9c9 12221 ℕ0cn0 12415 ;cdc 12621 ↑cexp 13998 logb clogb 26747 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-inf2 9564 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118 ax-addf 11119 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-fi 9328 df-sup 9359 df-inf 9360 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13279 df-ioc 13280 df-ico 13281 df-icc 13282 df-fz 13438 df-fzo 13585 df-fl 13726 df-mod 13804 df-seq 13939 df-exp 13999 df-fac 14211 df-bc 14240 df-hash 14268 df-shft 15004 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-limsup 15408 df-clim 15425 df-rlim 15426 df-sum 15624 df-ef 16004 df-sin 16006 df-cos 16007 df-pi 16009 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-rest 17356 df-topn 17357 df-0g 17375 df-gsum 17376 df-topgen 17377 df-pt 17378 df-prds 17381 df-xrs 17437 df-qtop 17442 df-imas 17443 df-xps 17445 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-mulg 19015 df-cntz 19263 df-cmn 19728 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-fbas 21323 df-fg 21324 df-cnfld 21327 df-top 22855 df-topon 22872 df-topsp 22894 df-bases 22907 df-cld 22980 df-ntr 22981 df-cls 22982 df-nei 23059 df-lp 23097 df-perf 23098 df-cn 23188 df-cnp 23189 df-haus 23276 df-tx 23523 df-hmeo 23716 df-fil 23807 df-fm 23899 df-flim 23900 df-flf 23901 df-xms 24281 df-ms 24282 df-tms 24283 df-cncf 24844 df-limc 25840 df-dv 25841 df-log 26538 df-cxp 26539 df-logb 26748 |
| This theorem is referenced by: 3lexlogpow5ineq3 42456 aks4d1lem1 42461 aks4d1p1 42475 aks4d1p6 42480 aks4d1p7d1 42481 aks4d1p7 42482 aks4d1p8 42486 |
| Copyright terms: Public domain | W3C validator |