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Mirrors > Home > ILE Home > Th. List > 2irrexpq | Unicode version |
Description: There exist real numbers
and which are not rational such
that is
rational. Statement in the Metamath book, section
1.1.5, footnote 27 on page 17, and the "constructive proof"
for theorem
1.2 of [Bauer], p. 483. This is a
constructive proof because it is
based on two explicitly named non-rational numbers and
logb , see sqrt2irr0 12039, 2logb9irr 13299 and
sqrt2cxp2logb9e3 13303. Therefore, this proof is acceptable/usable
in
intuitionistic logic.
For a theorem which is the same but proves that and are irrational (in the sense of being apart from any rational number), see 2irrexpqap 13306. (Contributed by AV, 23-Dec-2022.) |
Ref | Expression |
---|---|
2irrexpq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrt2irr0 12039 | . 2 | |
2 | 2logb9irr 13299 | . 2 logb | |
3 | sqrt2cxp2logb9e3 13303 | . . 3 logb | |
4 | 3z 9196 | . . . 4 | |
5 | zq 9535 | . . . 4 | |
6 | 4, 5 | ax-mp 5 | . . 3 |
7 | 3, 6 | eqeltri 2230 | . 2 logb |
8 | oveq1 5831 | . . . 4 | |
9 | 8 | eleq1d 2226 | . . 3 |
10 | oveq2 5832 | . . . 4 logb logb | |
11 | 10 | eleq1d 2226 | . . 3 logb logb |
12 | 9, 11 | rspc2ev 2831 | . 2 logb logb |
13 | 1, 2, 7, 12 | mp3an 1319 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1335 wcel 2128 wrex 2436 cdif 3099 cfv 5170 (class class class)co 5824 cr 7731 c2 8884 c3 8885 c9 8891 cz 9167 cq 9528 csqrt 10896 ccxp 13189 logb clogb 13271 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 ax-pre-suploc 7853 ax-addf 7854 ax-mulf 7855 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-disj 3943 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-isom 5179 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-of 6032 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-frec 6338 df-1o 6363 df-2o 6364 df-oadd 6367 df-er 6480 df-map 6595 df-pm 6596 df-en 6686 df-dom 6687 df-fin 6688 df-sup 6928 df-inf 6929 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-5 8895 df-6 8896 df-7 8897 df-8 8898 df-9 8899 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-xneg 9679 df-xadd 9680 df-ioo 9796 df-ico 9798 df-icc 9799 df-fz 9913 df-fzo 10042 df-fl 10169 df-mod 10222 df-seqfrec 10345 df-exp 10419 df-fac 10600 df-bc 10622 df-ihash 10650 df-shft 10715 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 df-sumdc 11251 df-ef 11545 df-e 11546 df-dvds 11684 df-gcd 11830 df-prm 11985 df-rest 12364 df-topgen 12383 df-psmet 12398 df-xmet 12399 df-met 12400 df-bl 12401 df-mopn 12402 df-top 12407 df-topon 12420 df-bases 12452 df-ntr 12507 df-cn 12599 df-cnp 12600 df-tx 12664 df-cncf 12969 df-limced 13036 df-dvap 13037 df-relog 13190 df-rpcxp 13191 df-logb 13272 |
This theorem is referenced by: (None) |
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