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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 2982 | . . 3 | |
2 | qre 8710 | . . . . . . 7 | |
3 | remulcl 7101 | . . . . . . 7 | |
4 | 2, 3 | sylan2 280 | . . . . . 6 |
5 | 4 | ad2ant2r 492 | . . . . 5 |
6 | qdivcl 8728 | . . . . . . . . . . . . 13 | |
7 | 6 | 3expb 1139 | . . . . . . . . . . . 12 |
8 | 7 | expcom 114 | . . . . . . . . . . 11 |
9 | 8 | adantl 271 | . . . . . . . . . 10 |
10 | recn 7106 | . . . . . . . . . . . . . 14 | |
11 | 10 | 3ad2ant1 959 | . . . . . . . . . . . . 13 |
12 | qcn 8719 | . . . . . . . . . . . . . 14 | |
13 | 12 | 3ad2ant2 960 | . . . . . . . . . . . . 13 |
14 | simp3 940 | . . . . . . . . . . . . . 14 | |
15 | 0z 8362 | . . . . . . . . . . . . . . . . 17 | |
16 | zq 8711 | . . . . . . . . . . . . . . . . 17 | |
17 | 15, 16 | ax-mp 7 | . . . . . . . . . . . . . . . 16 |
18 | qapne 8724 | . . . . . . . . . . . . . . . 16 # | |
19 | 17, 18 | mpan2 415 | . . . . . . . . . . . . . . 15 # |
20 | 19 | 3ad2ant2 960 | . . . . . . . . . . . . . 14 # |
21 | 14, 20 | mpbird 165 | . . . . . . . . . . . . 13 # |
22 | 11, 13, 21 | divcanap4d 7883 | . . . . . . . . . . . 12 |
23 | 22 | 3expb 1139 | . . . . . . . . . . 11 |
24 | 23 | eleq1d 2147 | . . . . . . . . . 10 |
25 | 9, 24 | sylibd 147 | . . . . . . . . 9 |
26 | 25 | con3d 593 | . . . . . . . 8 |
27 | 26 | ex 113 | . . . . . . 7 |
28 | 27 | com23 77 | . . . . . 6 |
29 | 28 | imp31 252 | . . . . 5 |
30 | 5, 29 | jca 300 | . . . 4 |
31 | 30 | 3impb 1134 | . . 3 |
32 | 1, 31 | syl3an1b 1205 | . 2 |
33 | eldif 2982 | . 2 | |
34 | 32, 33 | sylibr 132 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wne 2245 cdif 2970 class class class wbr 3785 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 cmul 6986 # cap 7681 cdiv 7760 cz 8351 cq 8704 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-q 8705 |
This theorem is referenced by: (None) |
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