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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 12333 | . . 3 ⊢ 9 ∈ ℝ | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 9 ∈ ℝ) |
| 3 | 1nn0 12510 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 4 | 1nn 12245 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
| 5 | 3, 4 | decnncl 12719 | . . . . . 6 ⊢ ;11 ∈ ℕ |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → ;11 ∈ ℕ) |
| 7 | 6 | nnred 12249 | . . . 4 ⊢ (𝜑 → ;11 ∈ ℝ) |
| 8 | 7re 12327 | . . . . 5 ⊢ 7 ∈ ℝ | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 7 ∈ ℝ) |
| 10 | 0red 11239 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 11 | 7pos 12345 | . . . . . . 7 ⊢ 0 < 7 | |
| 12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 7) |
| 13 | 10, 12 | ltned 11372 | . . . . 5 ⊢ (𝜑 → 0 ≠ 7) |
| 14 | 13 | necomd 2991 | . . . 4 ⊢ (𝜑 → 7 ≠ 0) |
| 15 | 7, 9, 14 | redivcld 12064 | . . 3 ⊢ (𝜑 → (;11 / 7) ∈ ℝ) |
| 16 | 5nn0 12514 | . . . 4 ⊢ 5 ∈ ℕ0 | |
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 5 ∈ ℕ0) |
| 18 | 15, 17 | reexpcld 14151 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ∈ ℝ) |
| 19 | 2re 12308 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
| 21 | 2pos 12337 | . . . . 5 ⊢ 0 < 2 | |
| 22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 < 2) |
| 23 | 3lexlogpow5ineq4.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 24 | 3re 12314 | . . . . . 6 ⊢ 3 ∈ ℝ | |
| 25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → 3 ∈ ℝ) |
| 26 | 3pos 12339 | . . . . . 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 11396 | . . . 4 ⊢ (𝜑 → 0 < 𝑋) |
| 30 | 1red 11237 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 31 | 1lt2 12405 | . . . . . . 7 ⊢ 1 < 2 | |
| 32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 < 2) |
| 33 | 30, 32 | ltned 11372 | . . . . 5 ⊢ (𝜑 → 1 ≠ 2) |
| 34 | 33 | necomd 2991 | . . . 4 ⊢ (𝜑 → 2 ≠ 1) |
| 35 | 20, 22, 23, 29, 34 | relogbcld 41380 | . . 3 ⊢ (𝜑 → (2 logb 𝑋) ∈ ℝ) |
| 36 | 35, 17 | reexpcld 14151 | . 2 ⊢ (𝜑 → ((2 logb 𝑋)↑5) ∈ ℝ) |
| 37 | 3lexlogpow5ineq1 41462 | . . 3 ⊢ 9 < ((;11 / 7)↑5) | |
| 38 | 37 | a1i 11 | . 2 ⊢ (𝜑 → 9 < ((;11 / 7)↑5)) |
| 39 | 23, 28 | 3lexlogpow5ineq2 41463 | . 2 ⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) |
| 40 | 2, 18, 36, 38, 39 | ltletrd 11396 | 1 ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℝcr 11129 0cc0 11130 1c1 11131 < clt 11270 ≤ cle 11271 / cdiv 11893 ℕcn 12234 2c2 12289 3c3 12290 5c5 12292 7c7 12294 9c9 12296 ℕ0cn0 12494 ;cdc 12699 ↑cexp 14050 logb clogb 26683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-fi 9426 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ioc 13353 df-ico 13354 df-icc 13355 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-fac 14257 df-bc 14286 df-hash 14314 df-shft 15038 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 df-rlim 15457 df-sum 15657 df-ef 16035 df-sin 16037 df-cos 16038 df-pi 16040 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-rest 17395 df-topn 17396 df-0g 17414 df-gsum 17415 df-topgen 17416 df-pt 17417 df-prds 17420 df-xrs 17475 df-qtop 17480 df-imas 17481 df-xps 17483 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-mulg 19015 df-cntz 19259 df-cmn 19728 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-fbas 21263 df-fg 21264 df-cnfld 21267 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-cld 22910 df-ntr 22911 df-cls 22912 df-nei 22989 df-lp 23027 df-perf 23028 df-cn 23118 df-cnp 23119 df-haus 23206 df-tx 23453 df-hmeo 23646 df-fil 23737 df-fm 23829 df-flim 23830 df-flf 23831 df-xms 24213 df-ms 24214 df-tms 24215 df-cncf 24785 df-limc 25782 df-dv 25783 df-log 26477 df-cxp 26478 df-logb 26684 |
| This theorem is referenced by: 3lexlogpow5ineq3 41465 aks4d1lem1 41470 aks4d1p1 41484 aks4d1p6 41489 aks4d1p7d1 41490 aks4d1p7 41491 aks4d1p8 41495 |
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