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Theorem irrmul 10341
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 3162 . . 3  |-  ( A  e.  ( RR  \  QQ )  <->  ( A  e.  RR  /\  -.  A  e.  QQ ) )
2 qre 10321 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
3 remulcl 8822 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
42, 3sylan2 460 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  QQ )  ->  ( A  x.  B
)  e.  RR )
54ad2ant2r 727 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  RR )
6 qdivcl 10337 . . . . . . . . . . . . 13  |-  ( ( ( A  x.  B
)  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  e.  QQ )
763expb 1152 . . . . . . . . . . . 12  |-  ( ( ( A  x.  B
)  e.  QQ  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  e.  QQ )
87expcom 424 . . . . . . . . . . 11  |-  ( ( B  e.  QQ  /\  B  =/=  0 )  -> 
( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B
)  e.  QQ ) )
98adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B )  e.  QQ ) )
10 qcn 10330 . . . . . . . . . . . . 13  |-  ( B  e.  QQ  ->  B  e.  CC )
11 recn 8827 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  CC )
12 divcan4 9449 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
1311, 12syl3an1 1215 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
1410, 13syl3an2 1216 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
15143expb 1152 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  =  A )
1615eleq1d 2349 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( (
( A  x.  B
)  /  B )  e.  QQ  <->  A  e.  QQ ) )
179, 16sylibd 205 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  A  e.  QQ ) )
1817con3d 125 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B
)  e.  QQ ) )
1918ex 423 . . . . . . 7  |-  ( A  e.  RR  ->  (
( B  e.  QQ  /\  B  =/=  0 )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B )  e.  QQ ) ) )
2019com23 72 . . . . . 6  |-  ( A  e.  RR  ->  ( -.  A  e.  QQ  ->  ( ( B  e.  QQ  /\  B  =/=  0 )  ->  -.  ( A  x.  B
)  e.  QQ ) ) )
2120imp31 421 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  -.  ( A  x.  B
)  e.  QQ )
225, 21jca 518 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  RR  /\ 
-.  ( A  x.  B )  e.  QQ ) )
23223impb 1147 . . 3  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
241, 23syl3an1b 1218 . 2  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  e.  RR  /\  -.  ( A  x.  B
)  e.  QQ ) )
25 eldif 3162 . 2  |-  ( ( A  x.  B )  e.  ( RR  \  QQ )  <->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
2624, 25sylibr 203 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    / cdiv 9423   QQcq 10316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-q 10317
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