Theorem prodgt0 8468

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 504	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> B e. RR )
2	1	renegcld 8009	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u B e. RR )
3		simplll 503	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> A e. RR )
4	3	renegcld 8009	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u A e. RR )
5		simplr 500	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. RR )
6	5	lt0neg1d 8144	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( B < 0 <-> 0 < -u B ) )
7	6	biimpa 292	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u B )
8		simprr 502	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( A x. B ) )
9		simpll 499	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. RR )
10	9	recnd 7666	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. CC )
11	5	recnd 7666	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. CC )
12	10, 11	mul2negd 8042	. . . . . . . . 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 3901	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u A x. -u B ) )
14	10	negcld 7931	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u A e. CC )
15	11	negcld 7931	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u B e. CC )
16	14, 15	mulcomd 7659	. . . . . . . 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 3899	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u B x. -u A ) )
18	17	adantr 272	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < ( -u B x. -u A ) )
19		prodgt0gt0 8467	. . . . . 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 1185	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u A )
21	3	lt0neg1d 8144	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> ( A < 0 <-> 0 < -u A ) )
22	20, 21	mpbird 166	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> A < 0 )
23		simplrl 505	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 <_ A )
24		0red 7639	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 e. RR )
25	24, 3	lenltd 7751	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> ( 0 <_ A <-> -. A < 0 ) )
26	23, 25	mpbid 146	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -. A < 0 )
27	22, 26	pm2.65da 628	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -. B < 0 )
28		0red 7639	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. RR )
29	28, 5	lenltd 7751	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( 0 <_ B <-> -. B < 0 ) )
30	27, 29	mpbird 166	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 <_ B )
31	9, 5	remulcld 7668	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. RR )
32	31, 8	gt0ap0d 8257	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) # 0 )
33	10, 11, 32	mulap0bbd 8282	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B # 0 )
34		0cnd 7631	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. CC )
35		apsym 8234	. . . 4 |- ( ( B e. CC /\ 0 e. CC ) -> ( B # 0 <-> 0 # B ) )
36	11, 34, 35	syl2anc 406	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( B # 0 <-> 0 # B ) )
37	33, 36	mpbid 146	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 # B )
38		ltleap 8259	. . 3 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> ( 0 <_ B /\ 0 # B ) ) )
39	28, 5, 38	syl2anc 406	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( 0 < B <-> ( 0 <_ B /\ 0 # B ) ) )
40	30, 37, 39	mpbir2and 896	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1448   class class class wbr 3875  (class class class)co 5706   CCcc 7498   RRcr 7499   0cc0 7500   x. cmul 7505   < clt 7672   <_ cle 7673   -ucneg 7805   # cap 8209

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294

This theorem is referenced by:  prodgt02  8469  prodgt0i  8524  evennn2n  11375