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Theorem irrmul 9645
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 3138 . . 3  |-  ( A  e.  ( RR  \  QQ )  <->  ( A  e.  RR  /\  -.  A  e.  QQ ) )
2 qre 9623 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
3 remulcl 7938 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
42, 3sylan2 286 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  QQ )  ->  ( A  x.  B
)  e.  RR )
54ad2ant2r 509 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  RR )
6 qdivcl 9641 . . . . . . . . . . . . 13  |-  ( ( ( A  x.  B
)  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  e.  QQ )
763expb 1204 . . . . . . . . . . . 12  |-  ( ( ( A  x.  B
)  e.  QQ  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  e.  QQ )
87expcom 116 . . . . . . . . . . 11  |-  ( ( B  e.  QQ  /\  B  =/=  0 )  -> 
( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B
)  e.  QQ ) )
98adantl 277 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B )  e.  QQ ) )
10 recn 7943 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  CC )
11103ad2ant1 1018 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  A  e.  CC )
12 qcn 9632 . . . . . . . . . . . . . 14  |-  ( B  e.  QQ  ->  B  e.  CC )
13123ad2ant2 1019 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  B  e.  CC )
14 simp3 999 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  B  =/=  0 )
15 0z 9262 . . . . . . . . . . . . . . . . 17  |-  0  e.  ZZ
16 zq 9624 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
1715, 16ax-mp 5 . . . . . . . . . . . . . . . 16  |-  0  e.  QQ
18 qapne 9637 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  QQ  /\  0  e.  QQ )  ->  ( B #  0  <->  B  =/=  0 ) )
1917, 18mpan2 425 . . . . . . . . . . . . . . 15  |-  ( B  e.  QQ  ->  ( B #  0  <->  B  =/=  0
) )
20193ad2ant2 1019 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( B #  0  <->  B  =/=  0
) )
2114, 20mpbird 167 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  B #  0 )
2211, 13, 21divcanap4d 8751 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
23223expb 1204 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  =  A )
2423eleq1d 2246 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( (
( A  x.  B
)  /  B )  e.  QQ  <->  A  e.  QQ ) )
259, 24sylibd 149 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  A  e.  QQ ) )
2625con3d 631 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B
)  e.  QQ ) )
2726ex 115 . . . . . . 7  |-  ( A  e.  RR  ->  (
( B  e.  QQ  /\  B  =/=  0 )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B )  e.  QQ ) ) )
2827com23 78 . . . . . 6  |-  ( A  e.  RR  ->  ( -.  A  e.  QQ  ->  ( ( B  e.  QQ  /\  B  =/=  0 )  ->  -.  ( A  x.  B
)  e.  QQ ) ) )
2928imp31 256 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  -.  ( A  x.  B
)  e.  QQ )
305, 29jca 306 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  RR  /\ 
-.  ( A  x.  B )  e.  QQ ) )
31303impb 1199 . . 3  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
321, 31syl3an1b 1274 . 2  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  e.  RR  /\  -.  ( A  x.  B
)  e.  QQ ) )
33 eldif 3138 . 2  |-  ( ( A  x.  B )  e.  ( RR  \  QQ )  <->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
3432, 33sylibr 134 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347    \ cdif 3126   class class class wbr 4003  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810    x. cmul 7815   # cap 8536    / cdiv 8627   ZZcz 9251   QQcq 9617
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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