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Theorem reapmul1 8325
Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8516. (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 7734 . . . . 5  |-  0  e.  RR
2 reaplt 8318 . . . . 5  |-  ( ( C  e.  RR  /\  0  e.  RR )  ->  ( C #  0  <->  ( C  <  0  \/  0  <  C ) ) )
31, 2mpan2 421 . . . 4  |-  ( C  e.  RR  ->  ( C #  0  <->  ( C  <  0  \/  0  < 
C ) ) )
43pm5.32i 449 . . 3  |-  ( ( C  e.  RR  /\  C #  0 )  <->  ( C  e.  RR  /\  ( C  <  0  \/  0  <  C ) ) )
5 simp1 966 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  A  e.  RR )
65recnd 7762 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  A  e.  CC )
7 simp3l 994 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  e.  RR )
87recnd 7762 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  e.  CC )
96, 8mulneg2d 8142 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A  x.  -u C
)  =  -u ( A  x.  C )
)
10 simp2 967 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  B  e.  RR )
1110recnd 7762 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  B  e.  CC )
1211, 8mulneg2d 8142 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( B  x.  -u C
)  =  -u ( B  x.  C )
)
139, 12breq12d 3912 . . . . . . . 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 8110 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u C  e.  RR )
15 simp3r 995 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  C  <  0 )
167lt0neg1d 8245 . . . . . . . . . 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 8324 . . . . . . . . 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 1205 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  -u C ) #  ( B  x.  -u C
) ) )
205, 7remulcld 7764 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A  x.  C
)  e.  RR )
2110, 7remulcld 7764 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( B  x.  C
)  e.  RR )
2220, 21ltnegd 8253 . . . . . . . . . 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 8253 . . . . . . . . . 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 767 . . . . . . . . 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 8318 . . . . . . . . . 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 408 . . . . . . . . 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 8110 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u ( A  x.  C
)  e.  RR )
2821renegcld 8110 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C  <  0 ) )  ->  -u ( B  x.  C
)  e.  RR )
29 reaplt 8318 . . . . . . . . . . 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 408 . . . . . . . . . 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 702 . . . . . . . . . 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 1166 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  C  <  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
3635anassrs 397 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  C  <  0 )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
37 reapmul1lem 8324 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
38373expa 1166 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
3938anassrs 397 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  0  <  C )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
4036, 39jaodan 771 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  /\  ( C  <  0  \/  0  <  C ) )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C
) ) )
4140anasss 396 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  ( C  <  0  \/  0  <  C ) ) )  ->  ( A #  B 
<->  ( A  x.  C
) #  ( B  x.  C ) ) )
424, 41sylan2b 285 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  C #  0 ) )  ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
43423impa 1161 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 682    /\ w3a 947    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588    x. cmul 7593    < clt 7768   -ucneg 7902   # cap 8311
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-ltxr 7773  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312
This theorem is referenced by: (None)
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