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Theorem remulext1 8757
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 1021 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2 simp3 1023 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
31, 2remulcld 8188 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. C ) e. RR )
4 simp2 1022 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
54, 2remulcld 8188 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B x. C ) e. RR )
6 reaplt 8746 . . 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 411 . 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 8128 . . . 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 8223 . . . . 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 411 . . . 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 8746 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
121, 4, 11syl2anc 411 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
13 ltxrlt 8223 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
141, 4, 13syl2anc 411 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
15 ltxrlt 8223 . . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> B <RR A ) )
164, 1, 15syl2anc 411 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < A <-> B <RR A ) )
1714, 16orbi12d 798 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B \/ B < A ) <-> ( A <RR B \/ B <RR A ) ) )
1812, 17bitrd 188 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B <-> ( A <RR B \/ B <RR A ) ) )
198, 10, 183imtr4d 203 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) < ( B x. C ) -> A # B ) )
20 ax-pre-mulext 8128 . . . . 5 |- ( ( B e. RR /\ A e. RR /\ C e. RR ) -> ( ( B x. C ) <RR ( A x. C ) -> ( B <RR A \/ A <RR B ) ) )
21203com12 1231 . . . 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 8223 . . . . 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 411 . . . 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 733 . . . . 5 |- ( ( A <RR B \/ B <RR A ) <-> ( B <RR A \/ A <RR B ) )
2518, 24bitrdi 196 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B <-> ( B <RR A \/ A <RR B ) ) )
2621, 23, 253imtr4d 203 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B x. C ) < ( A x. C ) -> A # B ) )
2719, 26jaod 722 . 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 150 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 105   \/ wo 713   /\ w3a 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   RRcr 8009   <RR cltrr 8014   x. cmul 8015   < clt 8192   # cap 8739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740
This theorem is referenced by:  remulext2  8758  mulext1  8770
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