Theorem prodgt0 8811

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 534	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> B e. RR )
2	1	renegcld 8339	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u B e. RR )
3		simplll 533	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> A e. RR )
4	3	renegcld 8339	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u A e. RR )
5		simplr 528	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. RR )
6	5	lt0neg1d 8474	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( B < 0 <-> 0 < -u B ) )
7	6	biimpa 296	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u B )
8		simprr 531	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( A x. B ) )
9		simpll 527	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. RR )
10	9	recnd 7988	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. CC )
11	5	recnd 7988	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. CC )
12	10, 11	mul2negd 8372	. . . . . . . . 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 4033	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u A x. -u B ) )
14	10	negcld 8257	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u A e. CC )
15	11	negcld 8257	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u B e. CC )
16	14, 15	mulcomd 7981	. . . . . . . 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 4031	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u B x. -u A ) )
18	17	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < ( -u B x. -u A ) )
19		prodgt0gt0 8810	. . . . . 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 1239	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u A )
21	3	lt0neg1d 8474	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> ( A < 0 <-> 0 < -u A ) )
22	20, 21	mpbird 167	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> A < 0 )
23		simplrl 535	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 <_ A )
24		0red 7960	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 e. RR )
25	24, 3	lenltd 8077	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> ( 0 <_ A <-> -. A < 0 ) )
26	23, 25	mpbid 147	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -. A < 0 )
27	22, 26	pm2.65da 661	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -. B < 0 )
28		0red 7960	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. RR )
29	28, 5	lenltd 8077	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( 0 <_ B <-> -. B < 0 ) )
30	27, 29	mpbird 167	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 <_ B )
31	9, 5	remulcld 7990	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. RR )
32	31, 8	gt0ap0d 8588	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) # 0 )
33	10, 11, 32	mulap0bbd 8619	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B # 0 )
34		0cnd 7952	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. CC )
35		apsym 8565	. . . 4 |- ( ( B e. CC /\ 0 e. CC ) -> ( B # 0 <-> 0 # B ) )
36	11, 34, 35	syl2anc 411	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( B # 0 <-> 0 # B ) )
37	33, 36	mpbid 147	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 # B )
38		ltleap 8591	. . 3 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> ( 0 <_ B /\ 0 # B ) ) )
39	28, 5, 38	syl2anc 411	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( 0 < B <-> ( 0 <_ B /\ 0 # B ) ) )
40	30, 37, 39	mpbir2and 944	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   x. cmul 7818   < clt 7994   <_ cle 7995   -ucneg 8131   # cap 8540

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632

This theorem is referenced by:  prodgt02  8812  prodgt0i  8867  evennn2n  11890