Theorem reapmul1 8742

Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8935. (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 8146	. . . . 5 |- 0 e. RR
2		reaplt 8735	. . . . 5 |- ( ( C e. RR /\ 0 e. RR ) -> ( C # 0 <-> ( C < 0 \/ 0 < C ) ) )
3	1, 2	mpan2 425	. . . 4 |- ( C e. RR -> ( C # 0 <-> ( C < 0 \/ 0 < C ) ) )
4	3	pm5.32i 454	. . 3 |- ( ( C e. RR /\ C # 0 ) <-> ( C e. RR /\ ( C < 0 \/ 0 < C ) ) )
5		simp1 1021	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. RR )
6	5	recnd 8175	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. CC )
7		simp3l 1049	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. RR )
8	7	recnd 8175	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. CC )
9	6, 8	mulneg2d 8558	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. -u C ) = -u ( A x. C ) )
10		simp2 1022	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. RR )
11	10	recnd 8175	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. CC )
12	11, 8	mulneg2d 8558	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. -u C ) = -u ( B x. C ) )
13	9, 12	breq12d 4096	. . . . . . . 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 8526	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u C e. RR )
15		simp3r 1050	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C < 0 )
16	7	lt0neg1d 8662	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( C < 0 <-> 0 < -u C ) )
17	15, 16	mpbid 147	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> 0 < -u C )
18		reapmul1lem 8741	. . . . . . . . 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 1275	. . . . . . . 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 8177	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. C ) e. RR )
21	10, 7	remulcld 8177	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. C ) e. RR )
22	20, 21	ltnegd 8670	. . . . . . . . . 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 8670	. . . . . . . . . 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 798	. . . . . . . . 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 8735	. . . . . . . . . 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 411	. . . . . . . . 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 8526	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( A x. C ) e. RR )
28	21	renegcld 8526	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( B x. C ) e. RR )
29		reaplt 8735	. . . . . . . . . . 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 411	. . . . . . . . . 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 733	. . . . . . . . . 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	bitrdi 196	. . . . . . . . 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 220	. . . . . . . 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 220	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
35	34	3expa 1227	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ C < 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
36	35	anassrs 400	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) /\ C < 0 ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
37		reapmul1lem 8741	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
38	37	3expa 1227	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
39	38	anassrs 400	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) /\ 0 < C ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
40	36, 39	jaodan 802	. . . 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 399	. . 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 287	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
43	42	3impa 1218	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 104   <-> wb 105   \/ wo 713   /\ w3a 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   x. cmul 8004   < clt 8181   -ucneg 8318   # cap 8728

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 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

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 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729

This theorem is referenced by: (None)