Theorem reapmul1 7586

Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7764. (Contributed by Jim Kingdon, 8-Feb-2020.)

Assertion

Ref	Expression
reapmul1	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )

Proof of Theorem reapmul1

Step	Hyp	Ref	Expression
1		0re 7027	. . . . 5 |- 0 e. RR
2		reaplt 7579	. . . . 5 |- ( ( C e. RR /\ 0 e. RR ) -> ( C # 0 <-> ( C < 0 \/ 0 < C ) ) )
3	1, 2	mpan2 401	. . . 4 |- ( C e. RR -> ( C # 0 <-> ( C < 0 \/ 0 < C ) ) )
4	3	pm5.32i 427	. . 3 |- ( ( C e. RR /\ C # 0 ) <-> ( C e. RR /\ ( C < 0 \/ 0 < C ) ) )
5		simp1 904	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. RR )
6	5	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. CC )
7		simp3l 932	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. RR )
8	7	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. CC )
9	6, 8	mulneg2d 7409	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. -u C ) = -u ( A x. C ) )
10		simp2 905	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. RR )
11	10	recnd 7054	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. CC )
12	11, 8	mulneg2d 7409	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. -u C ) = -u ( B x. C ) )
13	9, 12	breq12d 3777	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( ( A x. -u C ) # ( B x. -u C ) <-> -u ( A x. C ) # -u ( B x. C ) ) )
14	7	renegcld 7378	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u C e. RR )
15		simp3r 933	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C < 0 )
16	7	lt0neg1d 7507	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( C < 0 <-> 0 < -u C ) )
17	15, 16	mpbid 135	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> 0 < -u C )
18		reapmul1lem 7585	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( -u C e. RR /\ 0 < -u C ) ) -> ( A # B <-> ( A x. -u C ) # ( B x. -u C ) ) )
19	5, 10, 14, 17, 18	syl112anc 1139	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A # B <-> ( A x. -u C ) # ( B x. -u C ) ) )
20	5, 7	remulcld 7056	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. C ) e. RR )
21	10, 7	remulcld 7056	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. C ) e. RR )
22	20, 21	ltnegd 7514	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( ( A x. C ) < ( B x. C ) <-> -u ( B x. C ) < -u ( A x. C ) ) )
23	21, 20	ltnegd 7514	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( ( B x. C ) < ( A x. C ) <-> -u ( A x. C ) < -u ( B x. C ) ) )
24	22, 23	orbi12d 707	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( ( ( A x. C ) < ( B x. C ) \/ ( B x. C ) < ( A x. C ) ) <-> ( -u ( B x. C ) < -u ( A x. C ) \/ -u ( A x. C ) < -u ( B x. C ) ) ) )
25		reaplt 7579	. . . . . . . . . 10 |- ( ( ( 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 ) ) ) )
26	20, 21, 25	syl2anc 391	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( ( A x. C ) # ( B x. C ) <-> ( ( A x. C ) < ( B x. C ) \/ ( B x. C ) < ( A x. C ) ) ) )
27	20	renegcld 7378	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( A x. C ) e. RR )
28	21	renegcld 7378	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( B x. C ) e. RR )
29		reaplt 7579	. . . . . . . . . . 11 |- ( ( -u ( A x. C ) e. RR /\ -u ( B x. C ) e. RR ) -> ( -u ( A x. C ) # -u ( B x. C ) <-> ( -u ( A x. C ) < -u ( B x. C ) \/ -u ( B x. C ) < -u ( A x. C ) ) ) )
30	27, 28, 29	syl2anc 391	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( -u ( A x. C ) # -u ( B x. C ) <-> ( -u ( A x. C ) < -u ( B x. C ) \/ -u ( B x. C ) < -u ( A x. C ) ) ) )
31		orcom 647	. . . . . . . . . 10 |- ( ( -u ( A x. C ) < -u ( B x. C ) \/ -u ( B x. C ) < -u ( A x. C ) ) <-> ( -u ( B x. C ) < -u ( A x. C ) \/ -u ( A x. C ) < -u ( B x. C ) ) )
32	30, 31	syl6bb 185	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( -u ( A x. C ) # -u ( B x. C ) <-> ( -u ( B x. C ) < -u ( A x. C ) \/ -u ( A x. C ) < -u ( B x. C ) ) ) )
33	24, 26, 32	3bitr4d 209	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( ( A x. C ) # ( B x. C ) <-> -u ( A x. C ) # -u ( B x. C ) ) )
34	13, 19, 33	3bitr4d 209	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
35	34	3expa 1104	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ C < 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
36	35	anassrs 380	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) /\ C < 0 ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
37		reapmul1lem 7585	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
38	37	3expa 1104	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
39	38	anassrs 380	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) /\ 0 < C ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
40	36, 39	jaodan 710	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) /\ ( C < 0 \/ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
41	40	anasss 379	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ ( C < 0 \/ 0 < C ) ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
42	4, 41	sylan2b 271	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
43	42	3impa 1099	1 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888   0cc0 6889   x. cmul 6894   < clt 7060   -ucneg 7183   # 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

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: (None)