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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 12388 | . . 3 ⊢ 9 ∈ ℝ | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 9 ∈ ℝ) |
| 3 | 1nn0 12565 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 4 | 1nn 12300 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
| 5 | 3, 4 | decnncl 12774 | . . . . . 6 ⊢ ;11 ∈ ℕ |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → ;11 ∈ ℕ) |
| 7 | 6 | nnred 12304 | . . . 4 ⊢ (𝜑 → ;11 ∈ ℝ) |
| 8 | 7re 12382 | . . . . 5 ⊢ 7 ∈ ℝ | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 7 ∈ ℝ) |
| 10 | 0red 11289 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 11 | 7pos 12400 | . . . . . . 7 ⊢ 0 < 7 | |
| 12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 7) |
| 13 | 10, 12 | ltned 11422 | . . . . 5 ⊢ (𝜑 → 0 ≠ 7) |
| 14 | 13 | necomd 2998 | . . . 4 ⊢ (𝜑 → 7 ≠ 0) |
| 15 | 7, 9, 14 | redivcld 12118 | . . 3 ⊢ (𝜑 → (;11 / 7) ∈ ℝ) |
| 16 | 5nn0 12569 | . . . 4 ⊢ 5 ∈ ℕ0 | |
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 5 ∈ ℕ0) |
| 18 | 15, 17 | reexpcld 14209 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ∈ ℝ) |
| 19 | 2re 12363 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
| 21 | 2pos 12392 | . . . . 5 ⊢ 0 < 2 | |
| 22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 < 2) |
| 23 | 3lexlogpow5ineq4.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 24 | 3re 12369 | . . . . . 6 ⊢ 3 ∈ ℝ | |
| 25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → 3 ∈ ℝ) |
| 26 | 3pos 12394 | . . . . . 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 11446 | . . . 4 ⊢ (𝜑 → 0 < 𝑋) |
| 30 | 1red 11287 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 31 | 1lt2 12460 | . . . . . . 7 ⊢ 1 < 2 | |
| 32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 < 2) |
| 33 | 30, 32 | ltned 11422 | . . . . 5 ⊢ (𝜑 → 1 ≠ 2) |
| 34 | 33 | necomd 2998 | . . . 4 ⊢ (𝜑 → 2 ≠ 1) |
| 35 | 20, 22, 23, 29, 34 | relogbcld 41878 | . . 3 ⊢ (𝜑 → (2 logb 𝑋) ∈ ℝ) |
| 36 | 35, 17 | reexpcld 14209 | . 2 ⊢ (𝜑 → ((2 logb 𝑋)↑5) ∈ ℝ) |
| 37 | 3lexlogpow5ineq1 41960 | . . 3 ⊢ 9 < ((;11 / 7)↑5) | |
| 38 | 37 | a1i 11 | . 2 ⊢ (𝜑 → 9 < ((;11 / 7)↑5)) |
| 39 | 23, 28 | 3lexlogpow5ineq2 41961 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) |
| 40 | 2, 18, 36, 38, 39 | ltletrd 11446 | 1 ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2103 class class class wbr 5169 (class class class)co 7445 ℝcr 11179 0cc0 11180 1c1 11181 < clt 11320 ≤ cle 11321 / cdiv 11943 ℕcn 12289 2c2 12344 3c3 12345 5c5 12347 7c7 12349 9c9 12351 ℕ0cn0 12549 ;cdc 12754 ↑cexp 14108 logb clogb 26816 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ioc 13408 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-mod 13917 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 15731 df-ef 16109 df-sin 16111 df-cos 16112 df-pi 16114 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-mulg 19103 df-cntz 19352 df-cmn 19819 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-lp 23158 df-perf 23159 df-cn 23249 df-cnp 23250 df-haus 23337 df-tx 23584 df-hmeo 23777 df-fil 23868 df-fm 23960 df-flim 23961 df-flf 23962 df-xms 24344 df-ms 24345 df-tms 24346 df-cncf 24916 df-limc 25913 df-dv 25914 df-log 26607 df-cxp 26608 df-logb 26817 |
| This theorem is referenced by: 3lexlogpow5ineq3 41963 aks4d1lem1 41968 aks4d1p1 41982 aks4d1p6 41987 aks4d1p7d1 41988 aks4d1p7 41989 aks4d1p8 41993 |
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