Theorem prodgt0 7930

Description: Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
prodgt0	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < B )

Proof of Theorem prodgt0

Step	Hyp	Ref	Expression
1		simpllr 500	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> B e. RR )
2	1	renegcld 7484	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u B e. RR )
3		simplll 499	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> A e. RR )
4	3	renegcld 7484	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u A e. RR )
5		simplr 496	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. RR )
6	5	lt0neg1d 7616	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( B < 0 <-> 0 < -u B ) )
7	6	biimpa 290	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u B )
8		simprr 498	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( A x. B ) )
9		simpll 495	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. RR )
10	9	recnd 7147	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. CC )
11	5	recnd 7147	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. CC )
12	10, 11	mul2negd 7517	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( -u A x. -u B ) = ( A x. B ) )
13	8, 12	breqtrrd 3811	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u A x. -u B ) )
14	10	negcld 7406	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u A e. CC )
15	11	negcld 7406	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u B e. CC )
16	14, 15	mulcomd 7140	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( -u A x. -u B ) = ( -u B x. -u A ) )
17	13, 16	breqtrd 3809	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u B x. -u A ) )
18	17	adantr 270	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < ( -u B x. -u A ) )
19		prodgt0gt0 7929	. . . . . 6 |- ( ( ( -u B e. RR /\ -u A e. RR ) /\ ( 0 < -u B /\ 0 < ( -u B x. -u A ) ) ) -> 0 < -u A )
20	2, 4, 7, 18, 19	syl22anc 1170	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u A )
21	3	lt0neg1d 7616	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> ( A < 0 <-> 0 < -u A ) )
22	20, 21	mpbird 165	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> A < 0 )
23		simplrl 501	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 <_ A )
24		0red 7120	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 e. RR )
25	24, 3	lenltd 7227	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> ( 0 <_ A <-> -. A < 0 ) )
26	23, 25	mpbid 145	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -. A < 0 )
27	22, 26	pm2.65da 619	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -. B < 0 )
28		0red 7120	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. RR )
29	28, 5	lenltd 7227	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( 0 <_ B <-> -. B < 0 ) )
30	27, 29	mpbird 165	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 <_ B )
31	9, 5	remulcld 7149	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. RR )
32	31, 8	gt0ap0d 7728	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) # 0 )
33	10, 11, 32	mulap0bbd 7750	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B # 0 )
34		0cnd 7112	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. CC )
35		apsym 7706	. . . 4 |- ( ( B e. CC /\ 0 e. CC ) -> ( B # 0 <-> 0 # B ) )
36	11, 34, 35	syl2anc 403	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( B # 0 <-> 0 # B ) )
37	33, 36	mpbid 145	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 # B )
38		ltleap 7730	. . 3 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> ( 0 <_ B /\ 0 # B ) ) )
39	28, 5, 38	syl2anc 403	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( 0 < B <-> ( 0 <_ B /\ 0 # B ) ) )
40	30, 37, 39	mpbir2and 885	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   x. cmul 6986   < clt 7153   <_ cle 7154   -ucneg 7280   # cap 7681

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761

This theorem is referenced by:  prodgt02  7931  prodgt0i  7986  evennn2n  10283