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Theorem remulext1 8324
Description: Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
Assertion
Ref Expression
remulext1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
) #  ( B  x.  C )  ->  A #  B ) )

Proof of Theorem remulext1
StepHypRef Expression
1 simp1 964 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2 simp3 966 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
31, 2remulcld 7760 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
4 simp2 965 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
54, 2remulcld 7760 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C )  e.  RR )
6 reaplt 8313 . . 3  |-  ( ( ( 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 ) ) ) )
73, 5, 6syl2anc 406 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
) #  ( B  x.  C )  <->  ( ( A  x.  C )  <  ( B  x.  C
)  \/  ( B  x.  C )  < 
( A  x.  C
) ) ) )
8 ax-pre-mulext 7702 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <RR  ( B  x.  C )  ->  ( A  <RR  B  \/  B  <RR  A ) ) )
9 ltxrlt 7794 . . . . 5  |-  ( ( ( A  x.  C
)  e.  RR  /\  ( B  x.  C
)  e.  RR )  ->  ( ( A  x.  C )  < 
( B  x.  C
)  <->  ( A  x.  C )  <RR  ( B  x.  C ) ) )
103, 5, 9syl2anc 406 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( A  x.  C )  <RR  ( B  x.  C ) ) )
11 reaplt 8313 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  < 
A ) ) )
121, 4, 11syl2anc 406 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
13 ltxrlt 7794 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
141, 4, 13syl2anc 406 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
15 ltxrlt 7794 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
164, 1, 15syl2anc 406 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  <->  B  <RR  A ) )
1714, 16orbi12d 765 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  \/  B  <  A )  <-> 
( A  <RR  B  \/  B  <RR  A ) ) )
1812, 17bitrd 187 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  <RR  B  \/  B  <RR  A ) ) )
198, 10, 183imtr4d 202 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  ->  A #  B ) )
20 ax-pre-mulext 7702 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <RR  ( A  x.  C )  ->  ( B  <RR  A  \/  A  <RR  B ) ) )
21203com12 1168 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <RR  ( A  x.  C )  ->  ( B  <RR  A  \/  A  <RR  B ) ) )
22 ltxrlt 7794 . . . . 5  |-  ( ( ( B  x.  C
)  e.  RR  /\  ( A  x.  C
)  e.  RR )  ->  ( ( B  x.  C )  < 
( A  x.  C
)  <->  ( B  x.  C )  <RR  ( A  x.  C ) ) )
235, 3, 22syl2anc 406 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <  ( A  x.  C )  <->  ( B  x.  C )  <RR  ( A  x.  C ) ) )
24 orcom 700 . . . . 5  |-  ( ( A  <RR  B  \/  B  <RR  A )  <->  ( B  <RR  A  \/  A  <RR  B ) )
2518, 24syl6bb 195 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( B  <RR  A  \/  A  <RR  B ) ) )
2621, 23, 253imtr4d 202 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <  ( A  x.  C )  ->  A #  B ) )
2719, 26jaod 689 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( A  x.  C )  <  ( B  x.  C )  \/  ( B  x.  C
)  <  ( A  x.  C ) )  ->  A #  B ) )
287, 27sylbid 149 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
) #  ( B  x.  C )  ->  A #  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 680    /\ w3a 945    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583    <RR cltrr 7588    x. cmul 7589    < clt 7764   # 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  ax-pre-mulext 7702
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:  remulext2  8325  mulext1  8337
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