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Theorem remulext1 8354
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 981 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2 simp3 983 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
31, 2remulcld 7789 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. C ) e. RR )
4 simp2 982 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
54, 2remulcld 7789 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B x. C ) e. RR )
6 reaplt 8343 . . 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 408 . 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 7731 . . . 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 7823 . . . . 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 408 . . . 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 8343 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
121, 4, 11syl2anc 408 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
13 ltxrlt 7823 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
141, 4, 13syl2anc 408 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
15 ltxrlt 7823 . . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> B <RR A ) )
164, 1, 15syl2anc 408 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < A <-> B <RR A ) )
1714, 16orbi12d 782 . . . . 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 7731 . . . . 5 |- ( ( B e. RR /\ A e. RR /\ C e. RR ) -> ( ( B x. C ) <RR ( A x. C ) -> ( B <RR A \/ A <RR B ) ) )
21203com12 1185 . . . 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 7823 . . . . 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 408 . . . 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 717 . . . . 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 706 . 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 697   /\ w3a 962   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612   <RR cltrr 7617   x. cmul 7618   < clt 7793   # cap 8336
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-ltxr 7798  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337
This theorem is referenced by:  remulext2  8355  mulext1  8367
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