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| 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 12372 | . . 3 ⊢ 9 ∈ ℝ | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 9 ∈ ℝ) |
| 3 | 1nn0 12549 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 4 | 1nn 12284 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
| 5 | 3, 4 | decnncl 12760 | . . . . . 6 ⊢ ;11 ∈ ℕ |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → ;11 ∈ ℕ) |
| 7 | 6 | nnred 12288 | . . . 4 ⊢ (𝜑 → ;11 ∈ ℝ) |
| 8 | 7re 12366 | . . . . 5 ⊢ 7 ∈ ℝ | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 7 ∈ ℝ) |
| 10 | 0red 11271 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 11 | 7pos 12384 | . . . . . . 7 ⊢ 0 < 7 | |
| 12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 7) |
| 13 | 10, 12 | ltned 11404 | . . . . 5 ⊢ (𝜑 → 0 ≠ 7) |
| 14 | 13 | necomd 2996 | . . . 4 ⊢ (𝜑 → 7 ≠ 0) |
| 15 | 7, 9, 14 | redivcld 12102 | . . 3 ⊢ (𝜑 → (;11 / 7) ∈ ℝ) |
| 16 | 5nn0 12553 | . . . 4 ⊢ 5 ∈ ℕ0 | |
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 5 ∈ ℕ0) |
| 18 | 15, 17 | reexpcld 14209 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ∈ ℝ) |
| 19 | 2re 12347 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
| 21 | 2pos 12376 | . . . . 5 ⊢ 0 < 2 | |
| 22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 < 2) |
| 23 | 3lexlogpow5ineq4.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 24 | 3re 12353 | . . . . . 6 ⊢ 3 ∈ ℝ | |
| 25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → 3 ∈ ℝ) |
| 26 | 3pos 12378 | . . . . . 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 11428 | . . . 4 ⊢ (𝜑 → 0 < 𝑋) |
| 30 | 1red 11269 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 31 | 1lt2 12444 | . . . . . . 7 ⊢ 1 < 2 | |
| 32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 < 2) |
| 33 | 30, 32 | ltned 11404 | . . . . 5 ⊢ (𝜑 → 1 ≠ 2) |
| 34 | 33 | necomd 2996 | . . . 4 ⊢ (𝜑 → 2 ≠ 1) |
| 35 | 20, 22, 23, 29, 34 | relogbcld 41969 | . . 3 ⊢ (𝜑 → (2 logb 𝑋) ∈ ℝ) |
| 36 | 35, 17 | reexpcld 14209 | . 2 ⊢ (𝜑 → ((2 logb 𝑋)↑5) ∈ ℝ) |
| 37 | 3lexlogpow5ineq1 42050 | . . 3 ⊢ 9 < ((;11 / 7)↑5) | |
| 38 | 37 | a1i 11 | . 2 ⊢ (𝜑 → 9 < ((;11 / 7)↑5)) |
| 39 | 23, 28 | 3lexlogpow5ineq2 42051 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) |
| 40 | 2, 18, 36, 38, 39 | ltletrd 11428 | 1 ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5151 (class class class)co 7438 ℝcr 11161 0cc0 11162 1c1 11163 < clt 11302 ≤ cle 11303 / cdiv 11927 ℕcn 12273 2c2 12328 3c3 12329 5c5 12331 7c7 12333 9c9 12335 ℕ0cn0 12533 ;cdc 12740 ↑cexp 14108 logb clogb 26833 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 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-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-inf2 9688 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239 ax-pre-sup 11240 ax-addf 11241 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 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 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-se 5646 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-isom 6578 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-of 7704 df-om 7895 df-1st 8022 df-2nd 8023 df-supp 8194 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-2o 8515 df-er 8753 df-map 8876 df-pm 8877 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-fsupp 9409 df-fi 9458 df-sup 9489 df-inf 9490 df-oi 9557 df-card 9986 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-div 11928 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-q 12998 df-rp 13042 df-xneg 13161 df-xadd 13162 df-xmul 13163 df-ioo 13397 df-ioc 13398 df-ico 13399 df-icc 13400 df-fz 13554 df-fzo 13701 df-fl 13838 df-mod 13916 df-seq 14049 df-exp 14109 df-fac 14319 df-bc 14348 df-hash 14376 df-shft 15112 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-limsup 15513 df-clim 15530 df-rlim 15531 df-sum 15729 df-ef 16109 df-sin 16111 df-cos 16112 df-pi 16114 df-struct 17190 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-mulr 17321 df-starv 17322 df-sca 17323 df-vsca 17324 df-ip 17325 df-tset 17326 df-ple 17327 df-ds 17329 df-unif 17330 df-hom 17331 df-cco 17332 df-rest 17478 df-topn 17479 df-0g 17497 df-gsum 17498 df-topgen 17499 df-pt 17500 df-prds 17503 df-xrs 17558 df-qtop 17563 df-imas 17564 df-xps 17566 df-mre 17640 df-mrc 17641 df-acs 17643 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21383 df-xmet 21384 df-met 21385 df-bl 21386 df-mopn 21387 df-fbas 21388 df-fg 21389 df-cnfld 21392 df-top 22925 df-topon 22942 df-topsp 22964 df-bases 22978 df-cld 23052 df-ntr 23053 df-cls 23054 df-nei 23131 df-lp 23169 df-perf 23170 df-cn 23260 df-cnp 23261 df-haus 23348 df-tx 23595 df-hmeo 23788 df-fil 23879 df-fm 23971 df-flim 23972 df-flf 23973 df-xms 24355 df-ms 24356 df-tms 24357 df-cncf 24929 df-limc 25927 df-dv 25928 df-log 26624 df-cxp 26625 df-logb 26834 |
| This theorem is referenced by: 3lexlogpow5ineq3 42053 aks4d1lem1 42058 aks4d1p1 42072 aks4d1p6 42077 aks4d1p7d1 42078 aks4d1p7 42079 aks4d1p8 42083 |
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