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Mirrors > Home > ILE Home > Th. List > sqrt2irraplemnn | Unicode version |
Description: Lemma for sqrt2irrap 12174. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Ref | Expression |
---|---|
sqrt2irraplemnn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 |
. . . . . . 7
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2 | 1 | nnsqcld 10671 |
. . . . . 6
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3 | 2 | nnred 8930 |
. . . . 5
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4 | 0red 7957 |
. . . . . 6
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5 | 2 | nngt0d 8961 |
. . . . . 6
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6 | 4, 3, 5 | ltled 8074 |
. . . . 5
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7 | simpr 110 |
. . . . . . 7
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8 | 7 | nnsqcld 10671 |
. . . . . 6
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9 | 8 | nnrpd 9692 |
. . . . 5
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10 | 3, 6, 9 | sqrtdivd 11172 |
. . . 4
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11 | 1 | nnred 8930 |
. . . . . 6
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12 | 1 | nngt0d 8961 |
. . . . . . 7
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13 | 4, 11, 12 | ltled 8074 |
. . . . . 6
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14 | 11, 13 | sqrtsqd 11169 |
. . . . 5
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15 | 7 | nnred 8930 |
. . . . . 6
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16 | 7 | nngt0d 8961 |
. . . . . . 7
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17 | 4, 15, 16 | ltled 8074 |
. . . . . 6
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18 | 15, 17 | sqrtsqd 11169 |
. . . . 5
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19 | 14, 18 | oveq12d 5892 |
. . . 4
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20 | 10, 19 | eqtrd 2210 |
. . 3
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21 | sqne2sq 12171 |
. . . . . 6
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22 | 2 | nncnd 8931 |
. . . . . . . 8
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23 | 2cnd 8990 |
. . . . . . . 8
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24 | 8 | nncnd 8931 |
. . . . . . . 8
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25 | 8 | nnap0d 8963 |
. . . . . . . 8
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26 | 22, 23, 24, 25 | divmulap3d 8780 |
. . . . . . 7
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27 | 26 | necon3bid 2388 |
. . . . . 6
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28 | 21, 27 | mpbird 167 |
. . . . 5
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29 | 2 | nnzd 9372 |
. . . . . . 7
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30 | znq 9622 |
. . . . . . 7
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31 | 29, 8, 30 | syl2anc 411 |
. . . . . 6
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32 | 2z 9279 |
. . . . . . 7
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33 | zq 9624 |
. . . . . . 7
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34 | 32, 33 | mp1i 10 |
. . . . . 6
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35 | qapne 9637 |
. . . . . 6
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36 | 31, 34, 35 | syl2anc 411 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 28, 36 | mpbird 167 |
. . . 4
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38 | qre 9623 |
. . . . . 6
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39 | 31, 38 | syl 14 |
. . . . 5
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40 | 8 | nnred 8930 |
. . . . . . 7
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41 | 8 | nngt0d 8961 |
. . . . . . 7
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42 | 3, 40, 5, 41 | divgt0d 8890 |
. . . . . 6
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43 | 4, 39, 42 | ltled 8074 |
. . . . 5
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44 | 2re 8987 |
. . . . . 6
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45 | 44 | a1i 9 |
. . . . 5
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46 | 0le2 9007 |
. . . . . 6
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47 | 46 | a1i 9 |
. . . . 5
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48 | sqrt11ap 11042 |
. . . . 5
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49 | 39, 43, 45, 47, 48 | syl22anc 1239 |
. . . 4
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50 | 37, 49 | mpbird 167 |
. . 3
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51 | 20, 50 | eqbrtrrd 4027 |
. 2
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52 | nnz 9270 |
. . . . 5
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53 | znq 9622 |
. . . . 5
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54 | 52, 53 | sylan 283 |
. . . 4
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55 | qcn 9632 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
56 | 54, 55 | syl 14 |
. . 3
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57 | sqrt2re 12157 |
. . . . 5
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58 | 57 | recni 7968 |
. . . 4
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59 | 58 | a1i 9 |
. . 3
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60 | apsym 8561 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
61 | 56, 59, 60 | syl2anc 411 |
. 2
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62 | 51, 61 | mpbid 147 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-xor 1376 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-1o 6416 df-2o 6417 df-er 6534 df-en 6740 df-sup 6982 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-fz 10007 df-fzo 10140 df-fl 10267 df-mod 10320 df-seqfrec 10443 df-exp 10517 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-dvds 11790 df-gcd 11938 df-prm 12102 |
This theorem is referenced by: sqrt2irrap 12174 |
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