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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 3140 |
. . 3
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2 | qre 9627 |
. . . . . . 7
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3 | remulcl 7941 |
. . . . . . 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 9645 |
. . . . . . . . . . . . 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 7946 |
. . . . . . . . . . . . . 14
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11 | 10 | 3ad2ant1 1018 |
. . . . . . . . . . . . 13
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12 | qcn 9636 |
. . . . . . . . . . . . . 14
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13 | 12 | 3ad2ant2 1019 |
. . . . . . . . . . . . 13
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14 | simp3 999 |
. . . . . . . . . . . . . 14
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15 | 0z 9266 |
. . . . . . . . . . . . . . . . 17
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16 | zq 9628 |
. . . . . . . . . . . . . . . . 17
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17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
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18 | qapne 9641 |
. . . . . . . . . . . . . . . 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 8755 |
. . . . . . . . . . . 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 3140 |
. 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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 |
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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-n0 9179 df-z 9256 df-q 9622 |
This theorem is referenced by: 2logb9irrALT 14477 |
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