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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3lexlogpow5ineq3 | Structured version Visualization version GIF version |
Description: Combined inequality chain for a specific power of the binary logarithm, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.) |
Ref | Expression |
---|---|
3lexlogpow5ineq3.1 | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
3lexlogpow5ineq3.2 | ⊢ (𝜑 → 3 ≤ 𝑋) |
Ref | Expression |
---|---|
3lexlogpow5ineq3 | ⊢ (𝜑 → 7 < ((2 logb 𝑋)↑5)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7re 11920 | . . 3 ⊢ 7 ∈ ℝ | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 7 ∈ ℝ) |
3 | 9re 11926 | . . 3 ⊢ 9 ∈ ℝ | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 9 ∈ ℝ) |
5 | 2re 11901 | . . . . 5 ⊢ 2 ∈ ℝ | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ∈ ℝ) |
7 | 2pos 11930 | . . . . 5 ⊢ 0 < 2 | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 < 2) |
9 | 3lexlogpow5ineq3.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
10 | 0red 10833 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ) | |
11 | 3re 11907 | . . . . . 6 ⊢ 3 ∈ ℝ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → 3 ∈ ℝ) |
13 | 3pos 11932 | . . . . . 6 ⊢ 0 < 3 | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 < 3) |
15 | 3lexlogpow5ineq3.2 | . . . . 5 ⊢ (𝜑 → 3 ≤ 𝑋) | |
16 | 10, 12, 9, 14, 15 | ltletrd 10989 | . . . 4 ⊢ (𝜑 → 0 < 𝑋) |
17 | 1red 10831 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
18 | 1lt2 11998 | . . . . . . 7 ⊢ 1 < 2 | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 < 2) |
20 | 17, 19 | ltned 10965 | . . . . 5 ⊢ (𝜑 → 1 ≠ 2) |
21 | 20 | necomd 2994 | . . . 4 ⊢ (𝜑 → 2 ≠ 1) |
22 | 6, 8, 9, 16, 21 | relogbcld 39712 | . . 3 ⊢ (𝜑 → (2 logb 𝑋) ∈ ℝ) |
23 | 5nn0 12107 | . . . 4 ⊢ 5 ∈ ℕ0 | |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → 5 ∈ ℕ0) |
25 | 22, 24 | reexpcld 13730 | . 2 ⊢ (𝜑 → ((2 logb 𝑋)↑5) ∈ ℝ) |
26 | 7lt9 12027 | . . 3 ⊢ 7 < 9 | |
27 | 26 | a1i 11 | . 2 ⊢ (𝜑 → 7 < 9) |
28 | 9, 15 | 3lexlogpow5ineq4 39796 | . 2 ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) |
29 | 2, 4, 25, 27, 28 | lttrd 10990 | 1 ⊢ (𝜑 → 7 < ((2 logb 𝑋)↑5)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5050 (class class class)co 7210 ℝcr 10725 0cc0 10726 1c1 10727 < clt 10864 ≤ cle 10865 2c2 11882 3c3 11883 5c5 11885 7c7 11887 9c9 11889 ℕ0cn0 12087 ↑cexp 13632 logb clogb 25644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-inf2 9253 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 ax-addf 10805 ax-mulf 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-iin 4904 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-se 5507 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-isom 6386 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-of 7466 df-om 7642 df-1st 7758 df-2nd 7759 df-supp 7901 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-2o 8200 df-er 8388 df-map 8507 df-pm 8508 df-ixp 8576 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-fsupp 8983 df-fi 9024 df-sup 9055 df-inf 9056 df-oi 9123 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-uz 12436 df-q 12542 df-rp 12584 df-xneg 12701 df-xadd 12702 df-xmul 12703 df-ioo 12936 df-ioc 12937 df-ico 12938 df-icc 12939 df-fz 13093 df-fzo 13236 df-fl 13364 df-mod 13440 df-seq 13572 df-exp 13633 df-fac 13837 df-bc 13866 df-hash 13894 df-shft 14627 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-limsup 15029 df-clim 15046 df-rlim 15047 df-sum 15247 df-ef 15626 df-sin 15628 df-cos 15629 df-pi 15631 df-struct 16697 df-sets 16714 df-slot 16732 df-ndx 16742 df-base 16758 df-ress 16782 df-plusg 16812 df-mulr 16813 df-starv 16814 df-sca 16815 df-vsca 16816 df-ip 16817 df-tset 16818 df-ple 16819 df-ds 16821 df-unif 16822 df-hom 16823 df-cco 16824 df-rest 16924 df-topn 16925 df-0g 16943 df-gsum 16944 df-topgen 16945 df-pt 16946 df-prds 16949 df-xrs 17004 df-qtop 17009 df-imas 17010 df-xps 17012 df-mre 17086 df-mrc 17087 df-acs 17089 df-mgm 18111 df-sgrp 18160 df-mnd 18171 df-submnd 18216 df-mulg 18486 df-cntz 18708 df-cmn 19169 df-psmet 20352 df-xmet 20353 df-met 20354 df-bl 20355 df-mopn 20356 df-fbas 20357 df-fg 20358 df-cnfld 20361 df-top 21788 df-topon 21805 df-topsp 21827 df-bases 21840 df-cld 21913 df-ntr 21914 df-cls 21915 df-nei 21992 df-lp 22030 df-perf 22031 df-cn 22121 df-cnp 22122 df-haus 22209 df-tx 22456 df-hmeo 22649 df-fil 22740 df-fm 22832 df-flim 22833 df-flf 22834 df-xms 23215 df-ms 23216 df-tms 23217 df-cncf 23772 df-limc 24760 df-dv 24761 df-log 25442 df-cxp 25443 df-logb 25645 |
This theorem is referenced by: aks4d1p1p3 39808 aks4d1p1p2 39809 aks4d1p1p4 39810 aks4d1p2 39816 aks4d1p3 39817 |
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