Theorem reapmul1 8493

Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8684. (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 7899	. . . . 5 |- 0 e. RR
2		reaplt 8486	. . . . 5 |- ( ( C e. RR /\ 0 e. RR ) -> ( C # 0 <-> ( C < 0 \/ 0 < C ) ) )
3	1, 2	mpan2 422	. . . 4 |- ( C e. RR -> ( C # 0 <-> ( C < 0 \/ 0 < C ) ) )
4	3	pm5.32i 450	. . 3 |- ( ( C e. RR /\ C # 0 ) <-> ( C e. RR /\ ( C < 0 \/ 0 < C ) ) )
5		simp1 987	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. RR )
6	5	recnd 7927	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. CC )
7		simp3l 1015	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. RR )
8	7	recnd 7927	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. CC )
9	6, 8	mulneg2d 8310	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. -u C ) = -u ( A x. C ) )
10		simp2 988	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. RR )
11	10	recnd 7927	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. CC )
12	11, 8	mulneg2d 8310	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. -u C ) = -u ( B x. C ) )
13	9, 12	breq12d 3995	. . . . . . . 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 8278	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u C e. RR )
15		simp3r 1016	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C < 0 )
16	7	lt0neg1d 8413	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( C < 0 <-> 0 < -u C ) )
17	15, 16	mpbid 146	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> 0 < -u C )
18		reapmul1lem 8492	. . . . . . . . 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 1232	. . . . . . . 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 7929	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. C ) e. RR )
21	10, 7	remulcld 7929	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. C ) e. RR )
22	20, 21	ltnegd 8421	. . . . . . . . . 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 8421	. . . . . . . . . 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 783	. . . . . . . . 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 8486	. . . . . . . . . 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 409	. . . . . . . . 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 8278	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( A x. C ) e. RR )
28	21	renegcld 8278	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( B x. C ) e. RR )
29		reaplt 8486	. . . . . . . . . . 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 409	. . . . . . . . . 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 718	. . . . . . . . . 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 195	. . . . . . . . 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 219	. . . . . . . 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 219	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
35	34	3expa 1193	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ C < 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
36	35	anassrs 398	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) /\ C < 0 ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
37		reapmul1lem 8492	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
38	37	3expa 1193	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
39	38	anassrs 398	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) /\ 0 < C ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
40	36, 39	jaodan 787	. . . 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 397	. . 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 285	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
43	42	3impa 1184	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 103   <-> wb 104   \/ wo 698   /\ w3a 968   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   x. cmul 7758   < clt 7933   -ucneg 8070   # cap 8479

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480

This theorem is referenced by: (None)