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Theorem irrmul 10591
Description: The product of an irrational with a nonzero rational is irrational. (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 3322 . . 3  |-  ( A  e.  ( RR  \  QQ )  <->  ( A  e.  RR  /\  -.  A  e.  QQ ) )
2 qre 10571 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
3 remulcl 9067 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
42, 3sylan2 461 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  QQ )  ->  ( A  x.  B
)  e.  RR )
54ad2ant2r 728 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  RR )
6 qdivcl 10587 . . . . . . . . . . . . 13  |-  ( ( ( A  x.  B
)  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  e.  QQ )
763expb 1154 . . . . . . . . . . . 12  |-  ( ( ( A  x.  B
)  e.  QQ  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  e.  QQ )
87expcom 425 . . . . . . . . . . 11  |-  ( ( B  e.  QQ  /\  B  =/=  0 )  -> 
( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B
)  e.  QQ ) )
98adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B )  e.  QQ ) )
10 qcn 10580 . . . . . . . . . . . . 13  |-  ( B  e.  QQ  ->  B  e.  CC )
11 recn 9072 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  CC )
12 divcan4 9695 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
1311, 12syl3an1 1217 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
1410, 13syl3an2 1218 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
15143expb 1154 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  =  A )
1615eleq1d 2501 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( (
( A  x.  B
)  /  B )  e.  QQ  <->  A  e.  QQ ) )
179, 16sylibd 206 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  A  e.  QQ ) )
1817con3d 127 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B
)  e.  QQ ) )
1918ex 424 . . . . . . 7  |-  ( A  e.  RR  ->  (
( B  e.  QQ  /\  B  =/=  0 )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B )  e.  QQ ) ) )
2019com23 74 . . . . . 6  |-  ( A  e.  RR  ->  ( -.  A  e.  QQ  ->  ( ( B  e.  QQ  /\  B  =/=  0 )  ->  -.  ( A  x.  B
)  e.  QQ ) ) )
2120imp31 422 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  -.  ( A  x.  B
)  e.  QQ )
225, 21jca 519 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  RR  /\ 
-.  ( A  x.  B )  e.  QQ ) )
23223impb 1149 . . 3  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
241, 23syl3an1b 1220 . 2  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  e.  RR  /\  -.  ( A  x.  B
)  e.  QQ ) )
25 eldif 3322 . 2  |-  ( ( A  x.  B )  e.  ( RR  \  QQ )  <->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
2624, 25sylibr 204 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    / cdiv 9669   QQcq 10566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-q 10567
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