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Theorem remulext1 7590
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 904 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2 simp3 906 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
31, 2remulcld 7056 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
4 simp2 905 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
54, 2remulcld 7056 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C )  e.  RR )
6 reaplt 7579 . . 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 391 . 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 7002 . . . 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 7085 . . . . 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 391 . . . 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 7579 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  < 
A ) ) )
121, 4, 11syl2anc 391 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
13 ltxrlt 7085 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
141, 4, 13syl2anc 391 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
15 ltxrlt 7085 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
164, 1, 15syl2anc 391 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  <->  B  <RR  A ) )
1714, 16orbi12d 707 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  \/  B  <  A )  <-> 
( A  <RR  B  \/  B  <RR  A ) ) )
1812, 17bitrd 177 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  <RR  B  \/  B  <RR  A ) ) )
198, 10, 183imtr4d 192 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  ->  A #  B ) )
20 ax-pre-mulext 7002 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <RR  ( A  x.  C )  ->  ( B  <RR  A  \/  A  <RR  B ) ) )
21203com12 1108 . . . 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 7085 . . . . 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 391 . . . 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 647 . . . . 5  |-  ( ( A  <RR  B  \/  B  <RR  A )  <->  ( B  <RR  A  \/  A  <RR  B ) )
2518, 24syl6bb 185 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( B  <RR  A  \/  A  <RR  B ) ) )
2621, 23, 253imtr4d 192 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  x.  C
)  <  ( A  x.  C )  ->  A #  B ) )
2719, 26jaod 637 . 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 139 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 98    \/ wo 629    /\ w3a 885    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888    <RR cltrr 6893    x. cmul 6894    < clt 7060   # 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  ax-pre-mulext 7002
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:  remulext2  7591  mulext1  7603
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