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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 15607 resp. 2logb9irr 25376), satisfying the statement in 2irrexpqALT 25381. (Contributed by AV, 29-Dec-2022.) |
Ref | Expression |
---|---|
sqrt2cxp2logb9e3 | ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 11715 | . . . . . 6 ⊢ 2 ∈ ℂ | |
2 | cxpsqrt 25289 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑𝑐(1 / 2)) = (√‘2)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (2↑𝑐(1 / 2)) = (√‘2) |
4 | 3 | eqcomi 2833 | . . . 4 ⊢ (√‘2) = (2↑𝑐(1 / 2)) |
5 | 4 | oveq1i 7169 | . . 3 ⊢ ((√‘2)↑𝑐(2 logb 9)) = ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) |
6 | 2rp 12397 | . . . 4 ⊢ 2 ∈ ℝ+ | |
7 | halfre 11854 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
8 | 2z 12017 | . . . . . 6 ⊢ 2 ∈ ℤ | |
9 | uzid 12261 | . . . . . 6 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘2) |
11 | 9nn 11738 | . . . . . 6 ⊢ 9 ∈ ℕ | |
12 | nnrp 12403 | . . . . . 6 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ 9 ∈ ℝ+ |
14 | relogbzcl 25355 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 9 ∈ ℝ+) → (2 logb 9) ∈ ℝ) | |
15 | 10, 13, 14 | mp2an 690 | . . . 4 ⊢ (2 logb 9) ∈ ℝ |
16 | cxpcom 25323 | . . . 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 10658 | . . . . 5 ⊢ (2 logb 9) ∈ ℂ |
19 | cxpcl 25260 | . . . . 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 25289 | . . . 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 2851 | . 2 ⊢ ((√‘2)↑𝑐(2 logb 9)) = (√‘(2↑𝑐(2 logb 9))) |
24 | 2ne0 11744 | . . . . 5 ⊢ 2 ≠ 0 | |
25 | 1ne2 11848 | . . . . . 6 ⊢ 1 ≠ 2 | |
26 | 25 | necomi 3073 | . . . . 5 ⊢ 2 ≠ 1 |
27 | eldifpr 4600 | . . . . 5 ⊢ (2 ∈ (ℂ ∖ {0, 1}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1)) | |
28 | 1, 24, 26, 27 | mpbir3an 1337 | . . . 4 ⊢ 2 ∈ (ℂ ∖ {0, 1}) |
29 | 9cn 11740 | . . . . 5 ⊢ 9 ∈ ℂ | |
30 | 9re 11739 | . . . . . 6 ⊢ 9 ∈ ℝ | |
31 | 9pos 11753 | . . . . . 6 ⊢ 0 < 9 | |
32 | 30, 31 | gt0ne0ii 11179 | . . . . 5 ⊢ 9 ≠ 0 |
33 | eldifsn 4722 | . . . . 5 ⊢ (9 ∈ (ℂ ∖ {0}) ↔ (9 ∈ ℂ ∧ 9 ≠ 0)) | |
34 | 29, 32, 33 | mpbir2an 709 | . . . 4 ⊢ 9 ∈ (ℂ ∖ {0}) |
35 | cxplogb 25367 | . . . 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 6676 | . 2 ⊢ (√‘(2↑𝑐(2 logb 9))) = (√‘9) |
38 | sqrt9 14636 | . 2 ⊢ (√‘9) = 3 | |
39 | 23, 37, 38 | 3eqtri 2851 | 1 ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 {csn 4570 {cpr 4572 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 / cdiv 11300 ℕcn 11641 2c2 11695 3c3 11696 9c9 11702 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 √csqrt 14595 ↑𝑐ccxp 25142 logb clogb 25345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-cxp 25144 df-logb 25346 |
This theorem is referenced by: 2irrexpqALT 25381 |
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