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Theorem remulext1 8518
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 992 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2 simp3 994 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
31, 2remulcld 7950 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. C ) e. RR )
4 simp2 993 . . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
54, 2remulcld 7950 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B x. C ) e. RR )
6 reaplt 8507 . . 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 7892 . . . 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 7985 . . . . 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 8507 . . . . . 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 7985 . . . . . . 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 7985 . . . . . . 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 788 . . . . 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 7892 . . . . 5 |- ( ( B e. RR /\ A e. RR /\ C e. RR ) -> ( ( B x. C ) <RR ( A x. C ) -> ( B <RR A \/ A <RR B ) ) )
21203com12 1202 . . . 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 7985 . . . . 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 723 . . . . 5 |- ( ( A <RR B \/ B <RR A ) <-> ( B <RR A \/ A <RR B ) )
2518, 24bitrdi 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 712 . 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 703   /\ w3a 973   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   <RR cltrr 7778   x. cmul 7779   < clt 7954   # cap 8500
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501
This theorem is referenced by:  remulext2  8519  mulext1  8531
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