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Mirrors > Home > MPE Home > Th. List > sqrt2cxp2logb9e3 | Structured version Visualization version GIF version |
Description: The square root of two to the power of the logarithm of nine to base two is three. (√‘2) and (2 logb 9) are irrational numbers (see sqrt2irr0 16133 resp. 2logb9irr 26145), satisfying the statement in 2irrexpqALT 26150. (Contributed by AV, 29-Dec-2022.) |
Ref | Expression |
---|---|
sqrt2cxp2logb9e3 | ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 12228 | . . . . . 6 ⊢ 2 ∈ ℂ | |
2 | cxpsqrt 26058 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑𝑐(1 / 2)) = (√‘2)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (2↑𝑐(1 / 2)) = (√‘2) |
4 | 3 | eqcomi 2745 | . . . 4 ⊢ (√‘2) = (2↑𝑐(1 / 2)) |
5 | 4 | oveq1i 7367 | . . 3 ⊢ ((√‘2)↑𝑐(2 logb 9)) = ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) |
6 | 2rp 12920 | . . . 4 ⊢ 2 ∈ ℝ+ | |
7 | halfre 12367 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
8 | 2z 12535 | . . . . . 6 ⊢ 2 ∈ ℤ | |
9 | uzid 12778 | . . . . . 6 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘2) |
11 | 9nn 12251 | . . . . . 6 ⊢ 9 ∈ ℕ | |
12 | nnrp 12926 | . . . . . 6 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ 9 ∈ ℝ+ |
14 | relogbzcl 26124 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 9 ∈ ℝ+) → (2 logb 9) ∈ ℝ) | |
15 | 10, 13, 14 | mp2an 690 | . . . 4 ⊢ (2 logb 9) ∈ ℝ |
16 | cxpcom 26092 | . . . 4 ⊢ ((2 ∈ ℝ+ ∧ (1 / 2) ∈ ℝ ∧ (2 logb 9) ∈ ℝ) → ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) = ((2↑𝑐(2 logb 9))↑𝑐(1 / 2))) | |
17 | 6, 7, 15, 16 | mp3an 1461 | . . 3 ⊢ ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) = ((2↑𝑐(2 logb 9))↑𝑐(1 / 2)) |
18 | 15 | recni 11169 | . . . . 5 ⊢ (2 logb 9) ∈ ℂ |
19 | cxpcl 26029 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ (2 logb 9) ∈ ℂ) → (2↑𝑐(2 logb 9)) ∈ ℂ) | |
20 | 1, 18, 19 | mp2an 690 | . . . 4 ⊢ (2↑𝑐(2 logb 9)) ∈ ℂ |
21 | cxpsqrt 26058 | . . . 4 ⊢ ((2↑𝑐(2 logb 9)) ∈ ℂ → ((2↑𝑐(2 logb 9))↑𝑐(1 / 2)) = (√‘(2↑𝑐(2 logb 9)))) | |
22 | 20, 21 | ax-mp 5 | . . 3 ⊢ ((2↑𝑐(2 logb 9))↑𝑐(1 / 2)) = (√‘(2↑𝑐(2 logb 9))) |
23 | 5, 17, 22 | 3eqtri 2768 | . 2 ⊢ ((√‘2)↑𝑐(2 logb 9)) = (√‘(2↑𝑐(2 logb 9))) |
24 | 2ne0 12257 | . . . . 5 ⊢ 2 ≠ 0 | |
25 | 1ne2 12361 | . . . . . 6 ⊢ 1 ≠ 2 | |
26 | 25 | necomi 2998 | . . . . 5 ⊢ 2 ≠ 1 |
27 | eldifpr 4618 | . . . . 5 ⊢ (2 ∈ (ℂ ∖ {0, 1}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1)) | |
28 | 1, 24, 26, 27 | mpbir3an 1341 | . . . 4 ⊢ 2 ∈ (ℂ ∖ {0, 1}) |
29 | 9cn 12253 | . . . . 5 ⊢ 9 ∈ ℂ | |
30 | 9re 12252 | . . . . . 6 ⊢ 9 ∈ ℝ | |
31 | 9pos 12266 | . . . . . 6 ⊢ 0 < 9 | |
32 | 30, 31 | gt0ne0ii 11691 | . . . . 5 ⊢ 9 ≠ 0 |
33 | eldifsn 4747 | . . . . 5 ⊢ (9 ∈ (ℂ ∖ {0}) ↔ (9 ∈ ℂ ∧ 9 ≠ 0)) | |
34 | 29, 32, 33 | mpbir2an 709 | . . . 4 ⊢ 9 ∈ (ℂ ∖ {0}) |
35 | cxplogb 26136 | . . . 4 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 9 ∈ (ℂ ∖ {0})) → (2↑𝑐(2 logb 9)) = 9) | |
36 | 28, 34, 35 | mp2an 690 | . . 3 ⊢ (2↑𝑐(2 logb 9)) = 9 |
37 | 36 | fveq2i 6845 | . 2 ⊢ (√‘(2↑𝑐(2 logb 9))) = (√‘9) |
38 | sqrt9 15158 | . 2 ⊢ (√‘9) = 3 | |
39 | 23, 37, 38 | 3eqtri 2768 | 1 ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3907 {csn 4586 {cpr 4588 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 / cdiv 11812 ℕcn 12153 2c2 12208 3c3 12209 9c9 12215 ℤcz 12499 ℤ≥cuz 12763 ℝ+crp 12915 √csqrt 15118 ↑𝑐ccxp 25911 logb clogb 26114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-cxp 25913 df-logb 26115 |
This theorem is referenced by: 2irrexpqALT 26150 |
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