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Theorem irrmul 8679
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.)
Assertion
Ref Expression
irrmul  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  x.  B )  e.  ( RR  \  QQ ) )

Proof of Theorem irrmul
StepHypRef Expression
1 eldif 2955 . . 3  |-  ( A  e.  ( RR  \  QQ )  <->  ( A  e.  RR  /\  -.  A  e.  QQ ) )
2 qre 8657 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
3 remulcl 7067 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
42, 3sylan2 274 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  QQ )  ->  ( A  x.  B
)  e.  RR )
54ad2ant2r 486 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  RR )
6 qdivcl 8675 . . . . . . . . . . . . 13  |-  ( ( ( A  x.  B
)  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  e.  QQ )
763expb 1116 . . . . . . . . . . . 12  |-  ( ( ( A  x.  B
)  e.  QQ  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  e.  QQ )
87expcom 113 . . . . . . . . . . 11  |-  ( ( B  e.  QQ  /\  B  =/=  0 )  -> 
( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B
)  e.  QQ ) )
98adantl 266 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B )  e.  QQ ) )
10 recn 7072 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  CC )
11103ad2ant1 936 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  A  e.  CC )
12 qcn 8666 . . . . . . . . . . . . . 14  |-  ( B  e.  QQ  ->  B  e.  CC )
13123ad2ant2 937 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  B  e.  CC )
14 simp3 917 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  B  =/=  0 )
15 0z 8313 . . . . . . . . . . . . . . . . 17  |-  0  e.  ZZ
16 zq 8658 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
1715, 16ax-mp 7 . . . . . . . . . . . . . . . 16  |-  0  e.  QQ
18 qapne 8671 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  QQ  /\  0  e.  QQ )  ->  ( B #  0  <->  B  =/=  0 ) )
1917, 18mpan2 409 . . . . . . . . . . . . . . 15  |-  ( B  e.  QQ  ->  ( B #  0  <->  B  =/=  0
) )
20193ad2ant2 937 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( B #  0  <->  B  =/=  0
) )
2114, 20mpbird 160 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  B #  0 )
2211, 13, 21divcanap4d 7846 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
23223expb 1116 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  =  A )
2423eleq1d 2122 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( (
( A  x.  B
)  /  B )  e.  QQ  <->  A  e.  QQ ) )
259, 24sylibd 142 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  A  e.  QQ ) )
2625con3d 571 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B
)  e.  QQ ) )
2726ex 112 . . . . . . 7  |-  ( A  e.  RR  ->  (
( B  e.  QQ  /\  B  =/=  0 )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B )  e.  QQ ) ) )
2827com23 76 . . . . . 6  |-  ( A  e.  RR  ->  ( -.  A  e.  QQ  ->  ( ( B  e.  QQ  /\  B  =/=  0 )  ->  -.  ( A  x.  B
)  e.  QQ ) ) )
2928imp31 247 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  -.  ( A  x.  B
)  e.  QQ )
305, 29jca 294 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  RR  /\ 
-.  ( A  x.  B )  e.  QQ ) )
31303impb 1111 . . 3  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
321, 31syl3an1b 1182 . 2  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  e.  RR  /\  -.  ( A  x.  B
)  e.  QQ ) )
33 eldif 2955 . 2  |-  ( ( A  x.  B )  e.  ( RR  \  QQ )  <->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
3432, 33sylibr 141 1  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  x.  B )  e.  ( RR  \  QQ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409    =/= wne 2220    \ cdif 2942   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947    x. cmul 6952   # cap 7646    / cdiv 7725   ZZcz 8302   QQcq 8651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-q 8652
This theorem is referenced by: (None)
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