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Theorem reapmul1 8320
Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8508. (Contributed by Jim Kingdon, 8-Feb-2020.)
Assertion
Ref Expression
reapmul1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C #  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )

Proof of Theorem reapmul1
StepHypRef Expression
1 0re 7730 . . . . 5  |-  0  e.  RR
2 reaplt 8313 . . . . 5  |-  ( ( C  e.  RR  /\  0  e.  RR )  ->  ( C #  0  <->  ( C  <  0  \/  0  <  C ) ) )
31, 2mpan2 419 . . . 4  |-  ( C  e.  RR  ->  ( C #  0  <->  ( C  <  0  \/  0  < 
C ) ) )
43pm5.32i 447 . . 3  |-  ( ( C  e.  RR  /\  C #  0 )  <->  ( C  e.  RR  /\  ( C  <  0  \/  0  <  C ) ) )
5 simp1 964 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  A  e.  RR )
65recnd 7758 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  A  e.  CC )
7 simp3l 992 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  e.  RR )
87recnd 7758 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  e.  CC )
96, 8mulneg2d 8138 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A  x.  -u C
)  =  -u ( A  x.  C )
)
10 simp2 965 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  B  e.  RR )
1110recnd 7758 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  B  e.  CC )
1211, 8mulneg2d 8138 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( B  x.  -u C
)  =  -u ( B  x.  C )
)
139, 12breq12d 3910 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( ( A  x.  -u C ) #  ( B  x.  -u C )  <->  -u ( A  x.  C ) #  -u ( B  x.  C
) ) )
147renegcld 8106 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u C  e.  RR )
15 simp3r 993 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  <  0 )
167lt0neg1d 8241 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( C  <  0  <->  0  <  -u C ) )
1715, 16mpbid 146 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
0  <  -u C )
18 reapmul1lem 8319 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( -u C  e.  RR  /\  0  <  -u C ) )  ->  ( A #  B  <->  ( A  x.  -u C
) #  ( B  x.  -u C ) ) )
195, 10, 14, 17, 18syl112anc 1203 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  -u C ) #  ( B  x.  -u C
) ) )
205, 7remulcld 7760 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A  x.  C
)  e.  RR )
2110, 7remulcld 7760 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( B  x.  C
)  e.  RR )
2220, 21ltnegd 8248 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  <->  -u ( B  x.  C
)  <  -u ( A  x.  C ) ) )
2321, 20ltnegd 8248 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( ( B  x.  C )  <  ( A  x.  C )  <->  -u ( A  x.  C
)  <  -u ( B  x.  C ) ) )
2422, 23orbi12d 765 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( ( ( A  x.  C )  < 
( B  x.  C
)  \/  ( B  x.  C )  < 
( A  x.  C
) )  <->  ( -u ( B  x.  C )  <  -u ( A  x.  C )  \/  -u ( A  x.  C )  <  -u ( B  x.  C ) ) ) )
25 reaplt 8313 . . . . . . . . . 10  |-  ( ( ( A  x.  C
)  e.  RR  /\  ( B  x.  C
)  e.  RR )  ->  ( ( A  x.  C ) #  ( B  x.  C )  <-> 
( ( A  x.  C )  <  ( B  x.  C )  \/  ( B  x.  C
)  <  ( A  x.  C ) ) ) )
2620, 21, 25syl2anc 406 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( ( A  x.  C ) #  ( B  x.  C )  <->  ( ( A  x.  C )  <  ( B  x.  C
)  \/  ( B  x.  C )  < 
( A  x.  C
) ) ) )
2720renegcld 8106 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u ( A  x.  C
)  e.  RR )
2821renegcld 8106 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u ( B  x.  C
)  e.  RR )
29 reaplt 8313 . . . . . . . . . . 11  |-  ( (
-u ( A  x.  C )  e.  RR  /\  -u ( B  x.  C
)  e.  RR )  ->  ( -u ( A  x.  C ) #  -u ( B  x.  C
)  <->  ( -u ( A  x.  C )  <  -u ( B  x.  C )  \/  -u ( B  x.  C )  <  -u ( A  x.  C ) ) ) )
3027, 28, 29syl2anc 406 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( -u ( A  x.  C ) #  -u ( B  x.  C )  <->  ( -u ( A  x.  C )  <  -u ( B  x.  C )  \/  -u ( B  x.  C )  <  -u ( A  x.  C ) ) ) )
31 orcom 700 . . . . . . . . . 10  |-  ( (
-u ( A  x.  C )  <  -u ( B  x.  C )  \/  -u ( B  x.  C )  <  -u ( A  x.  C )
)  <->  ( -u ( B  x.  C )  <  -u ( A  x.  C )  \/  -u ( A  x.  C )  <  -u ( B  x.  C ) ) )
3230, 31syl6bb 195 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( -u ( A  x.  C ) #  -u ( B  x.  C )  <->  ( -u ( B  x.  C )  <  -u ( A  x.  C )  \/  -u ( A  x.  C )  <  -u ( B  x.  C ) ) ) )
3324, 26, 323bitr4d 219 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( ( A  x.  C ) #  ( B  x.  C )  <->  -u ( A  x.  C ) #  -u ( B  x.  C
) ) )
3413, 19, 333bitr4d 219 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
35343expa 1164 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
3635anassrs 395 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  C  <  0 )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
37 reapmul1lem 8319 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
38373expa 1164 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
3938anassrs 395 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  0  <  C )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
4036, 39jaodan 769 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  ( C  <  0  \/  0  <  C ) )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C
) ) )
4140anasss 394 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  ( C  <  0  \/  0  <  C ) ) )  ->  ( A #  B 
<->  ( A  x.  C
) #  ( B  x.  C ) ) )
424, 41sylan2b 283 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  C #  0 ) )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
43423impa 1159 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C #  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583   0cc0 7584    x. cmul 7589    < clt 7764   -ucneg 7898   # cap 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-ltxr 7769  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307
This theorem is referenced by: (None)
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