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Mirrors > Home > ILE Home > Th. List > sqrt2irrap | Unicode version |
Description: The square root of 2 is
irrational. That is, for any rational number,
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Ref | Expression |
---|---|
sqrt2irrap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 9264 |
. . 3
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2 | 1 | biimpi 119 |
. 2
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3 | simplrl 505 |
. . . . . . . . 9
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4 | 3 | adantr 272 |
. . . . . . . 8
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5 | simplrr 506 |
. . . . . . . . 9
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6 | 5 | adantr 272 |
. . . . . . . 8
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7 | znq 9266 |
. . . . . . . . 9
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8 | qre 9267 |
. . . . . . . . 9
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9 | 7, 8 | syl 14 |
. . . . . . . 8
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10 | 4, 6, 9 | syl2anc 406 |
. . . . . . 7
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11 | sqrt2re 11634 |
. . . . . . . 8
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12 | 11 | a1i 9 |
. . . . . . 7
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13 | 0red 7639 |
. . . . . . . 8
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14 | 4 | zcnd 9026 |
. . . . . . . . . 10
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15 | 6 | nncnd 8592 |
. . . . . . . . . 10
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16 | 6 | nnap0d 8624 |
. . . . . . . . . 10
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17 | 14, 15, 16 | divrecapd 8414 |
. . . . . . . . 9
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18 | 4 | zred 9025 |
. . . . . . . . . 10
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19 | 6 | nnrecred 8625 |
. . . . . . . . . 10
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20 | simpr 109 |
. . . . . . . . . 10
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21 | 1red 7653 |
. . . . . . . . . . 11
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22 | 6 | nnrpd 9329 |
. . . . . . . . . . 11
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23 | 0le1 8110 |
. . . . . . . . . . . 12
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24 | 23 | a1i 9 |
. . . . . . . . . . 11
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25 | 21, 22, 24 | divge0d 9371 |
. . . . . . . . . 10
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26 | mulle0r 8560 |
. . . . . . . . . 10
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27 | 18, 19, 20, 25, 26 | syl22anc 1185 |
. . . . . . . . 9
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28 | 17, 27 | eqbrtrd 3895 |
. . . . . . . 8
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29 | 2re 8648 |
. . . . . . . . . 10
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30 | 2pos 8669 |
. . . . . . . . . 10
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31 | 29, 30 | sqrtgt0ii 10743 |
. . . . . . . . 9
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32 | 31 | a1i 9 |
. . . . . . . 8
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33 | 10, 13, 12, 28, 32 | lelttrd 7758 |
. . . . . . 7
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34 | 10, 12, 33 | gtapd 8264 |
. . . . . 6
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35 | 3 | adantr 272 |
. . . . . . . 8
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36 | simpr 109 |
. . . . . . . 8
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37 | elnnz 8916 |
. . . . . . . 8
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38 | 35, 36, 37 | sylanbrc 411 |
. . . . . . 7
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39 | 5 | adantr 272 |
. . . . . . 7
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40 | sqrt2irraplemnn 11649 |
. . . . . . 7
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41 | 38, 39, 40 | syl2anc 406 |
. . . . . 6
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42 | 0z 8917 |
. . . . . . . . 9
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43 | zlelttric 8951 |
. . . . . . . . 9
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44 | 42, 43 | mpan2 419 |
. . . . . . . 8
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45 | 44 | ad2antrl 477 |
. . . . . . 7
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46 | 45 | adantr 272 |
. . . . . 6
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47 | 34, 41, 46 | mpjaodan 753 |
. . . . 5
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48 | simpr 109 |
. . . . 5
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49 | 47, 48 | breqtrrd 3901 |
. . . 4
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50 | 49 | ex 114 |
. . 3
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51 | 50 | rexlimdvva 2516 |
. 2
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52 | 2, 51 | mpd 13 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-xor 1322 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-1o 6243 df-2o 6244 df-er 6359 df-en 6565 df-sup 6786 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-q 9262 df-rp 9292 df-fz 9632 df-fzo 9761 df-fl 9884 df-mod 9937 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-dvds 11289 df-gcd 11431 df-prm 11582 |
This theorem is referenced by: (None) |
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