Theorem reapmul1 7379

Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7546. (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 6825	. . . . 5 |- 0 e. RR
2		reaplt 7372	. . . . 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 903	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> A e. RR)
6	5	recnd 6851	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> A e. CC)
7		simp3l 931	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> C e. RR)
8	7	recnd 6851	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> C e. CC)
9	6, 8	mulneg2d 7205	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> (A x. -u C) = -u(A x. C))
10		simp2 904	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> B e. RR)
11	10	recnd 6851	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> B e. CC)
12	11, 8	mulneg2d 7205	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> (B x. -u C) = -u(B x. C))
13	9, 12	breq12d 3768	. . . . . . . 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 7174	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> -u C e. RR)
15		simp3r 932	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> C < 0)
16	7	lt0neg1d 7302	. . . . . . . . . 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 7378	. . . . . . . . 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 1138	. . . . . . . 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 6853	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> (A x. C) e. RR)
21	10, 7	remulcld 6853	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> (B x. C) e. RR)
22	20, 21	ltnegd 7309	. . . . . . . . . 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 7309	. . . . . . . . . 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 706	. . . . . . . . 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 7372	. . . . . . . . . 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 7174	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> -u(A x. C) e. RR)
28	21	renegcld 7174	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ C < 0)) -> -u(B x. C) e. RR)
29		reaplt 7372	. . . . . . . . . . 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 646	. . . . . . . . . 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 1103	. . . . . 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 7378	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C)) -> (A # B <-> (A x. C) # (B x. C)))
38	37	3expa 1103	. . . . . 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 709	. . . 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 1098	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 628   /\ w3a 884   e. wcel 1390   class class class wbr 3755  (class class class)co 5455   RRcr 6710   0cc0 6711   x. cmul 6716   < clt 6857   -ucneg 6980   # cap 7365

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-ltxr 6862  df-sub 6981  df-neg 6982  df-reap 7359  df-ap 7366

This theorem is referenced by: (None)