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Theorem reapmul1 7586
Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7764. (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 7027 . . . . 5  |-  0  e.  RR
2 reaplt 7579 . . . . 5  |-  ( ( C  e.  RR  /\  0  e.  RR )  ->  ( C #  0  <->  ( C  <  0  \/  0  <  C ) ) )
31, 2mpan2 401 . . . 4  |-  ( C  e.  RR  ->  ( C #  0  <->  ( C  <  0  \/  0  < 
C ) ) )
43pm5.32i 427 . . 3  |-  ( ( C  e.  RR  /\  C #  0 )  <->  ( C  e.  RR  /\  ( C  <  0  \/  0  <  C ) ) )
5 simp1 904 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  A  e.  RR )
65recnd 7054 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  A  e.  CC )
7 simp3l 932 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  e.  RR )
87recnd 7054 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  e.  CC )
96, 8mulneg2d 7409 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A  x.  -u C
)  =  -u ( A  x.  C )
)
10 simp2 905 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  B  e.  RR )
1110recnd 7054 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  B  e.  CC )
1211, 8mulneg2d 7409 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( B  x.  -u C
)  =  -u ( B  x.  C )
)
139, 12breq12d 3777 . . . . . . . 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 7378 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u C  e.  RR )
15 simp3r 933 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  <  0 )
167lt0neg1d 7507 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( C  <  0  <->  0  <  -u C ) )
1715, 16mpbid 135 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
0  <  -u C )
18 reapmul1lem 7585 . . . . . . . . 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 1139 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  -u C ) #  ( B  x.  -u C
) ) )
205, 7remulcld 7056 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A  x.  C
)  e.  RR )
2110, 7remulcld 7056 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( B  x.  C
)  e.  RR )
2220, 21ltnegd 7514 . . . . . . . . . 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 7514 . . . . . . . . . 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 707 . . . . . . . . 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 7579 . . . . . . . . . 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 391 . . . . . . . . 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 7378 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u ( A  x.  C
)  e.  RR )
2821renegcld 7378 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u ( B  x.  C
)  e.  RR )
29 reaplt 7579 . . . . . . . . . . 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 391 . . . . . . . . . 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 647 . . . . . . . . . 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 185 . . . . . . . . 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 209 . . . . . . . 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 209 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
35343expa 1104 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
3635anassrs 380 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  C  <  0 )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
37 reapmul1lem 7585 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
38373expa 1104 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
3938anassrs 380 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  0  <  C )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
4036, 39jaodan 710 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  ( C  <  0  \/  0  <  C ) )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C
) ) )
4140anasss 379 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  ( C  <  0  \/  0  <  C ) ) )  ->  ( A #  B 
<->  ( A  x.  C
) #  ( B  x.  C ) ) )
424, 41sylan2b 271 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  C #  0 ) )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
43423impa 1099 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 97    <-> wb 98    \/ wo 629    /\ w3a 885    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888   0cc0 6889    x. cmul 6894    < clt 7060   -ucneg 7183   # cap 7572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-ltxr 7065  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573
This theorem is referenced by: (None)
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