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Theorem irrmul 11289
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 3452 . . 3  |-  ( A  e.  ( RR  \  QQ )  <->  ( A  e.  RR  /\  -.  A  e.  QQ ) )
2 qre 11269 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  RR )
3 remulcl 9623 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
42, 3sylan2 476 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  QQ )  ->  ( A  x.  B
)  e.  RR )
54ad2ant2r 751 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  RR )
6 qdivcl 11285 . . . . . . . . . . . . 13  |-  ( ( ( A  x.  B
)  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  e.  QQ )
763expb 1206 . . . . . . . . . . . 12  |-  ( ( ( A  x.  B
)  e.  QQ  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  e.  QQ )
87expcom 436 . . . . . . . . . . 11  |-  ( ( B  e.  QQ  /\  B  =/=  0 )  -> 
( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B
)  e.  QQ ) )
98adantl 467 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  ( ( A  x.  B )  /  B )  e.  QQ ) )
10 qcn 11278 . . . . . . . . . . . . 13  |-  ( B  e.  QQ  ->  B  e.  CC )
11 recn 9628 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  CC )
12 divcan4 10294 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
1311, 12syl3an1 1297 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
1410, 13syl3an2 1298 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
15143expb 1206 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  /  B )  =  A )
1615eleq1d 2498 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( (
( A  x.  B
)  /  B )  e.  QQ  <->  A  e.  QQ ) )
179, 16sylibd 217 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( ( A  x.  B )  e.  QQ  ->  A  e.  QQ ) )
1817con3d 138 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B
)  e.  QQ ) )
1918ex 435 . . . . . . 7  |-  ( A  e.  RR  ->  (
( B  e.  QQ  /\  B  =/=  0 )  ->  ( -.  A  e.  QQ  ->  -.  ( A  x.  B )  e.  QQ ) ) )
2019com23 81 . . . . . 6  |-  ( A  e.  RR  ->  ( -.  A  e.  QQ  ->  ( ( B  e.  QQ  /\  B  =/=  0 )  ->  -.  ( A  x.  B
)  e.  QQ ) ) )
2120imp31 433 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  -.  ( A  x.  B
)  e.  QQ )
225, 21jca 534 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  RR  /\ 
-.  ( A  x.  B )  e.  QQ ) )
23223impb 1201 . . 3  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
241, 23syl3an1b 1300 . 2  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  (
( A  x.  B
)  e.  RR  /\  -.  ( A  x.  B
)  e.  QQ ) )
25 eldif 3452 . 2  |-  ( ( A  x.  B )  e.  ( RR  \  QQ )  <->  ( ( A  x.  B )  e.  RR  /\  -.  ( A  x.  B )  e.  QQ ) )
2624, 25sylibr 215 1  |-  ( ( A  e.  ( RR 
\  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  x.  B )  e.  ( RR  \  QQ ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538    x. cmul 9543    / cdiv 10268   QQcq 11264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-n0 10870  df-z 10938  df-q 11265
This theorem is referenced by: (None)
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