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Mirrors > Home > ILE Home > Th. List > irrmul | Unicode version |
Description: The product of a real which is not rational with a nonzero rational is not rational. Note that by "not rational" we mean the negation of "is rational" (whereas "irrational" is often defined to mean apart from any rational number - given excluded middle these two definitions would be equivalent). (Contributed by NM, 7-Nov-2008.) |
Ref | Expression |
---|---|
irrmul |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3138 |
. . 3
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2 | qre 9623 |
. . . . . . 7
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3 | remulcl 7938 |
. . . . . . 7
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4 | 2, 3 | sylan2 286 |
. . . . . 6
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5 | 4 | ad2ant2r 509 |
. . . . 5
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6 | qdivcl 9641 |
. . . . . . . . . . . . 13
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7 | 6 | 3expb 1204 |
. . . . . . . . . . . 12
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8 | 7 | expcom 116 |
. . . . . . . . . . 11
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9 | 8 | adantl 277 |
. . . . . . . . . 10
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10 | recn 7943 |
. . . . . . . . . . . . . 14
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11 | 10 | 3ad2ant1 1018 |
. . . . . . . . . . . . 13
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12 | qcn 9632 |
. . . . . . . . . . . . . 14
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13 | 12 | 3ad2ant2 1019 |
. . . . . . . . . . . . 13
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14 | simp3 999 |
. . . . . . . . . . . . . 14
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15 | 0z 9262 |
. . . . . . . . . . . . . . . . 17
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16 | zq 9624 |
. . . . . . . . . . . . . . . . 17
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17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
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18 | qapne 9637 |
. . . . . . . . . . . . . . . 16
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19 | 17, 18 | mpan2 425 |
. . . . . . . . . . . . . . 15
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20 | 19 | 3ad2ant2 1019 |
. . . . . . . . . . . . . 14
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21 | 14, 20 | mpbird 167 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 11, 13, 21 | divcanap4d 8751 |
. . . . . . . . . . . 12
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23 | 22 | 3expb 1204 |
. . . . . . . . . . 11
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24 | 23 | eleq1d 2246 |
. . . . . . . . . 10
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25 | 9, 24 | sylibd 149 |
. . . . . . . . 9
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26 | 25 | con3d 631 |
. . . . . . . 8
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27 | 26 | ex 115 |
. . . . . . 7
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28 | 27 | com23 78 |
. . . . . 6
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29 | 28 | imp31 256 |
. . . . 5
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30 | 5, 29 | jca 306 |
. . . 4
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31 | 30 | 3impb 1199 |
. . 3
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32 | 1, 31 | syl3an1b 1274 |
. 2
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33 | eldif 3138 |
. 2
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34 | 32, 33 | sylibr 134 |
1
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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-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 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 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 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-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-id 4293 df-po 4296 df-iso 4297 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-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 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-n0 9175 df-z 9252 df-q 9618 |
This theorem is referenced by: 2logb9irrALT 14285 |
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