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Mirrors > Home > MPE Home > Th. List > 2logb9irrALT | Structured version Visualization version GIF version |
Description: Alternate proof of 2logb9irr 24973: The logarithm of nine to base two is irrational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2logb9irrALT | ⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sq3 13280 | . . . . 5 ⊢ (3↑2) = 9 | |
2 | 1 | eqcomi 2787 | . . . 4 ⊢ 9 = (3↑2) |
3 | 2 | oveq2i 6933 | . . 3 ⊢ (2 logb 9) = (2 logb (3↑2)) |
4 | 2cn 11450 | . . . . 5 ⊢ 2 ∈ ℂ | |
5 | 2ne0 11486 | . . . . 5 ⊢ 2 ≠ 0 | |
6 | 1ne2 11590 | . . . . . 6 ⊢ 1 ≠ 2 | |
7 | 6 | necomi 3023 | . . . . 5 ⊢ 2 ≠ 1 |
8 | eldifpr 4426 | . . . . 5 ⊢ (2 ∈ (ℂ ∖ {0, 1}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1)) | |
9 | 4, 5, 7, 8 | mpbir3an 1398 | . . . 4 ⊢ 2 ∈ (ℂ ∖ {0, 1}) |
10 | 3rp 12143 | . . . 4 ⊢ 3 ∈ ℝ+ | |
11 | 2z 11761 | . . . 4 ⊢ 2 ∈ ℤ | |
12 | relogbzexp 24954 | . . . 4 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 3 ∈ ℝ+ ∧ 2 ∈ ℤ) → (2 logb (3↑2)) = (2 · (2 logb 3))) | |
13 | 9, 10, 11, 12 | mp3an 1534 | . . 3 ⊢ (2 logb (3↑2)) = (2 · (2 logb 3)) |
14 | 3, 13 | eqtri 2802 | . 2 ⊢ (2 logb 9) = (2 · (2 logb 3)) |
15 | 3cn 11456 | . . . . . 6 ⊢ 3 ∈ ℂ | |
16 | 3ne0 11488 | . . . . . 6 ⊢ 3 ≠ 0 | |
17 | eldifsn 4550 | . . . . . 6 ⊢ (3 ∈ (ℂ ∖ {0}) ↔ (3 ∈ ℂ ∧ 3 ≠ 0)) | |
18 | 15, 16, 17 | mpbir2an 701 | . . . . 5 ⊢ 3 ∈ (ℂ ∖ {0}) |
19 | logbcl 24945 | . . . . 5 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 3 ∈ (ℂ ∖ {0})) → (2 logb 3) ∈ ℂ) | |
20 | 9, 18, 19 | mp2an 682 | . . . 4 ⊢ (2 logb 3) ∈ ℂ |
21 | 4, 20 | mulcomi 10385 | . . 3 ⊢ (2 · (2 logb 3)) = ((2 logb 3) · 2) |
22 | 2logb3irr 24975 | . . . 4 ⊢ (2 logb 3) ∈ (ℝ ∖ ℚ) | |
23 | zq 12101 | . . . . 5 ⊢ (2 ∈ ℤ → 2 ∈ ℚ) | |
24 | 11, 23 | ax-mp 5 | . . . 4 ⊢ 2 ∈ ℚ |
25 | irrmul 12121 | . . . 4 ⊢ (((2 logb 3) ∈ (ℝ ∖ ℚ) ∧ 2 ∈ ℚ ∧ 2 ≠ 0) → ((2 logb 3) · 2) ∈ (ℝ ∖ ℚ)) | |
26 | 22, 24, 5, 25 | mp3an 1534 | . . 3 ⊢ ((2 logb 3) · 2) ∈ (ℝ ∖ ℚ) |
27 | 21, 26 | eqeltri 2855 | . 2 ⊢ (2 · (2 logb 3)) ∈ (ℝ ∖ ℚ) |
28 | 14, 27 | eqeltri 2855 | 1 ⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 {csn 4398 {cpr 4400 (class class class)co 6922 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 · cmul 10277 2c2 11430 3c3 11431 9c9 11437 ℤcz 11728 ℚcq 12095 ℝ+crp 12137 ↑cexp 13178 logb clogb 24942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-sin 15202 df-cos 15203 df-pi 15205 df-dvds 15388 df-gcd 15623 df-prm 15791 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-cxp 24741 df-logb 24943 |
This theorem is referenced by: (None) |
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