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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 16239 resp. 2logb9irr 26792), satisfying the statement in 2irrexpqALT 26797. (Contributed by AV, 29-Dec-2022.) |
Ref | Expression |
---|---|
sqrt2cxp2logb9e3 | ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 12325 | . . . . . 6 ⊢ 2 ∈ ℂ | |
2 | cxpsqrt 26699 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑𝑐(1 / 2)) = (√‘2)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (2↑𝑐(1 / 2)) = (√‘2) |
4 | 3 | eqcomi 2734 | . . . 4 ⊢ (√‘2) = (2↑𝑐(1 / 2)) |
5 | 4 | oveq1i 7429 | . . 3 ⊢ ((√‘2)↑𝑐(2 logb 9)) = ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) |
6 | 2rp 13019 | . . . 4 ⊢ 2 ∈ ℝ+ | |
7 | halfre 12464 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
8 | 2z 12632 | . . . . . 6 ⊢ 2 ∈ ℤ | |
9 | uzid 12875 | . . . . . 6 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘2) |
11 | 9nn 12348 | . . . . . 6 ⊢ 9 ∈ ℕ | |
12 | nnrp 13025 | . . . . . 6 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ 9 ∈ ℝ+ |
14 | relogbzcl 26771 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 9 ∈ ℝ+) → (2 logb 9) ∈ ℝ) | |
15 | 10, 13, 14 | mp2an 690 | . . . 4 ⊢ (2 logb 9) ∈ ℝ |
16 | cxpcom 26735 | . . . 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 1457 | . . 3 ⊢ ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) = ((2↑𝑐(2 logb 9))↑𝑐(1 / 2)) |
18 | 15 | recni 11265 | . . . . 5 ⊢ (2 logb 9) ∈ ℂ |
19 | cxpcl 26670 | . . . . 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 26699 | . . . 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 2757 | . 2 ⊢ ((√‘2)↑𝑐(2 logb 9)) = (√‘(2↑𝑐(2 logb 9))) |
24 | 2ne0 12354 | . . . . 5 ⊢ 2 ≠ 0 | |
25 | 1ne2 12458 | . . . . . 6 ⊢ 1 ≠ 2 | |
26 | 25 | necomi 2984 | . . . . 5 ⊢ 2 ≠ 1 |
27 | eldifpr 4662 | . . . . 5 ⊢ (2 ∈ (ℂ ∖ {0, 1}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1)) | |
28 | 1, 24, 26, 27 | mpbir3an 1338 | . . . 4 ⊢ 2 ∈ (ℂ ∖ {0, 1}) |
29 | 9cn 12350 | . . . . 5 ⊢ 9 ∈ ℂ | |
30 | 9re 12349 | . . . . . 6 ⊢ 9 ∈ ℝ | |
31 | 9pos 12363 | . . . . . 6 ⊢ 0 < 9 | |
32 | 30, 31 | gt0ne0ii 11787 | . . . . 5 ⊢ 9 ≠ 0 |
33 | eldifsn 4792 | . . . . 5 ⊢ (9 ∈ (ℂ ∖ {0}) ↔ (9 ∈ ℂ ∧ 9 ≠ 0)) | |
34 | 29, 32, 33 | mpbir2an 709 | . . . 4 ⊢ 9 ∈ (ℂ ∖ {0}) |
35 | cxplogb 26783 | . . . 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 6899 | . 2 ⊢ (√‘(2↑𝑐(2 logb 9))) = (√‘9) |
38 | sqrt9 15264 | . 2 ⊢ (√‘9) = 3 | |
39 | 23, 37, 38 | 3eqtri 2757 | 1 ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3941 {csn 4630 {cpr 4632 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 ℝcr 11144 0cc0 11145 1c1 11146 / cdiv 11908 ℕcn 12250 2c2 12305 3c3 12306 9c9 12312 ℤcz 12596 ℤ≥cuz 12860 ℝ+crp 13014 √csqrt 15224 ↑𝑐ccxp 26551 logb clogb 26761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-mod 13876 df-seq 14008 df-exp 14068 df-fac 14277 df-bc 14306 df-hash 14334 df-shft 15058 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-limsup 15459 df-clim 15476 df-rlim 15477 df-sum 15677 df-ef 16055 df-sin 16057 df-cos 16058 df-pi 16060 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-hom 17276 df-cco 17277 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17503 df-qtop 17508 df-imas 17509 df-xps 17511 df-mre 17585 df-mrc 17586 df-acs 17588 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-submnd 18760 df-mulg 19048 df-cntz 19297 df-cmn 19766 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24287 df-ms 24288 df-tms 24289 df-cncf 24859 df-limc 25856 df-dv 25857 df-log 26552 df-cxp 26553 df-logb 26762 |
This theorem is referenced by: 2irrexpqALT 26797 |
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