Theorem reapmul1 8614

Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8807. (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 8019	. . . . 5 |- 0 e. RR
2		reaplt 8607	. . . . 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 999	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. RR )
6	5	recnd 8048	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> A e. CC )
7		simp3l 1027	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. RR )
8	7	recnd 8048	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C e. CC )
9	6, 8	mulneg2d 8431	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. -u C ) = -u ( A x. C ) )
10		simp2 1000	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. RR )
11	10	recnd 8048	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> B e. CC )
12	11, 8	mulneg2d 8431	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. -u C ) = -u ( B x. C ) )
13	9, 12	breq12d 4042	. . . . . . . 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 8399	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u C e. RR )
15		simp3r 1028	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> C < 0 )
16	7	lt0neg1d 8534	. . . . . . . . . 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 8613	. . . . . . . . 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 1253	. . . . . . . 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 8050	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( A x. C ) e. RR )
21	10, 7	remulcld 8050	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> ( B x. C ) e. RR )
22	20, 21	ltnegd 8542	. . . . . . . . . 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 8542	. . . . . . . . . 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 794	. . . . . . . . 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 8607	. . . . . . . . . 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 8399	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( A x. C ) e. RR )
28	21	renegcld 8399	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0 ) ) -> -u ( B x. C ) e. RR )
29		reaplt 8607	. . . . . . . . . . 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 729	. . . . . . . . . 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 1205	. . . . . 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 8613	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
38	37	3expa 1205	. . . . . 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 798	. . . 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 1196	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 709   /\ w3a 980   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   0cc0 7872   x. cmul 7877   < clt 8054   -ucneg 8191   # cap 8600

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601

This theorem is referenced by: (None)