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Mirrors > Home > ILE Home > Th. List > sqrt2cxp2logb9e3 | 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 not rational (see sqrt2irr0 12075 resp. 2logb9irr 13436), satisfying the statement in 2irrexpq 13441. (Contributed by AV, 29-Dec-2022.) |
Ref | Expression |
---|---|
sqrt2cxp2logb9e3 | ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2rp 9585 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
2 | rpcxpsqrt 13389 | . . . . . 6 ⊢ (2 ∈ ℝ+ → (2↑𝑐(1 / 2)) = (√‘2)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (2↑𝑐(1 / 2)) = (√‘2) |
4 | 3 | eqcomi 2168 | . . . 4 ⊢ (√‘2) = (2↑𝑐(1 / 2)) |
5 | 4 | oveq1i 5846 | . . 3 ⊢ ((√‘2)↑𝑐(2 logb 9)) = ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) |
6 | halfre 9061 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
7 | 2z 9210 | . . . . . 6 ⊢ 2 ∈ ℤ | |
8 | uzid 9471 | . . . . . 6 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘2) |
10 | 9nn 9016 | . . . . . 6 ⊢ 9 ∈ ℕ | |
11 | nnrp 9590 | . . . . . 6 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ 9 ∈ ℝ+ |
13 | relogbzcl 13417 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 9 ∈ ℝ+) → (2 logb 9) ∈ ℝ) | |
14 | 9, 12, 13 | mp2an 423 | . . . 4 ⊢ (2 logb 9) ∈ ℝ |
15 | cxpcom 13404 | . . . 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))) | |
16 | 1, 6, 14, 15 | mp3an 1326 | . . 3 ⊢ ((2↑𝑐(1 / 2))↑𝑐(2 logb 9)) = ((2↑𝑐(2 logb 9))↑𝑐(1 / 2)) |
17 | rpcxpcl 13371 | . . . . 5 ⊢ ((2 ∈ ℝ+ ∧ (2 logb 9) ∈ ℝ) → (2↑𝑐(2 logb 9)) ∈ ℝ+) | |
18 | 1, 14, 17 | mp2an 423 | . . . 4 ⊢ (2↑𝑐(2 logb 9)) ∈ ℝ+ |
19 | rpcxpsqrt 13389 | . . . 4 ⊢ ((2↑𝑐(2 logb 9)) ∈ ℝ+ → ((2↑𝑐(2 logb 9))↑𝑐(1 / 2)) = (√‘(2↑𝑐(2 logb 9)))) | |
20 | 18, 19 | ax-mp 5 | . . 3 ⊢ ((2↑𝑐(2 logb 9))↑𝑐(1 / 2)) = (√‘(2↑𝑐(2 logb 9))) |
21 | 5, 16, 20 | 3eqtri 2189 | . 2 ⊢ ((√‘2)↑𝑐(2 logb 9)) = (√‘(2↑𝑐(2 logb 9))) |
22 | 1re 7889 | . . . . 5 ⊢ 1 ∈ ℝ | |
23 | 2re 8918 | . . . . 5 ⊢ 2 ∈ ℝ | |
24 | 1lt2 9017 | . . . . 5 ⊢ 1 < 2 | |
25 | 22, 23, 24 | gtapii 8523 | . . . 4 ⊢ 2 # 1 |
26 | rpcxplogb 13429 | . . . 4 ⊢ ((2 ∈ ℝ+ ∧ 2 # 1 ∧ 9 ∈ ℝ+) → (2↑𝑐(2 logb 9)) = 9) | |
27 | 1, 25, 12, 26 | mp3an 1326 | . . 3 ⊢ (2↑𝑐(2 logb 9)) = 9 |
28 | 27 | fveq2i 5483 | . 2 ⊢ (√‘(2↑𝑐(2 logb 9))) = (√‘9) |
29 | sqrt9 10976 | . 2 ⊢ (√‘9) = 3 | |
30 | 21, 28, 29 | 3eqtri 2189 | 1 ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 1c1 7745 # cap 8470 / cdiv 8559 ℕcn 8848 2c2 8899 3c3 8900 9c9 8906 ℤcz 9182 ℤ≥cuz 9457 ℝ+crp 9580 √csqrt 10924 ↑𝑐ccxp 13325 logb clogb 13408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 ax-pre-suploc 7865 ax-addf 7866 ax-mulf 7867 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-disj 3954 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-of 6044 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-map 6607 df-pm 6608 df-en 6698 df-dom 6699 df-fin 6700 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-xneg 9699 df-xadd 9700 df-ioo 9819 df-ico 9821 df-icc 9822 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-fac 10628 df-bc 10650 df-ihash 10678 df-shft 10743 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 df-ef 11575 df-e 11576 df-rest 12500 df-topgen 12519 df-psmet 12534 df-xmet 12535 df-met 12536 df-bl 12537 df-mopn 12538 df-top 12543 df-topon 12556 df-bases 12588 df-ntr 12643 df-cn 12735 df-cnp 12736 df-tx 12800 df-cncf 13105 df-limced 13172 df-dvap 13173 df-relog 13326 df-rpcxp 13327 df-logb 13409 |
This theorem is referenced by: 2irrexpq 13441 2irrexpqap 13443 |
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