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Theorem remulext1 8385
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 982 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2 simp3 984 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
31, 2remulcld 7820 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. C ) e. RR )
4 simp2 983 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
54, 2remulcld 7820 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B x. C ) e. RR )
6 reaplt 8374 . . 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 409 . 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 7762 . . . 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 7854 . . . . 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 409 . . . 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 8374 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
121, 4, 11syl2anc 409 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
13 ltxrlt 7854 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
141, 4, 13syl2anc 409 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
15 ltxrlt 7854 . . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> B <RR A ) )
164, 1, 15syl2anc 409 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < A <-> B <RR A ) )
1714, 16orbi12d 783 . . . . 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 7762 . . . . 5 |- ( ( B e. RR /\ A e. RR /\ C e. RR ) -> ( ( B x. C ) <RR ( A x. C ) -> ( B <RR A \/ A <RR B ) ) )
21203com12 1186 . . . 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 7854 . . . . 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 409 . . . 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 718 . . . . 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 707 . 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 698   /\ w3a 963   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   RRcr 7643   <RR cltrr 7648   x. cmul 7649   < clt 7824   # cap 8367
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-ltxr 7829  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368
This theorem is referenced by:  remulext2  8386  mulext1  8398
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