Theorem prodgt0 8896

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 8423	. . . . . 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 8423	. . . . . 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 8559	. . . . . . 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 8072	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. CC )
11	5	recnd 8072	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. CC )
12	10, 11	mul2negd 8456	. . . . . . . . 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 4062	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u A x. -u B ) )
14	10	negcld 8341	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u A e. CC )
15	11	negcld 8341	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u B e. CC )
16	14, 15	mulcomd 8065	. . . . . . . 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 4060	. . . . . . 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 8895	. . . . . 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 1250	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u A )
21	3	lt0neg1d 8559	. . . . 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 8044	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 e. RR )
25	24, 3	lenltd 8161	. . . . 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 662	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -. B < 0 )
28		0red 8044	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. RR )
29	28, 5	lenltd 8161	. . 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 8074	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. RR )
32	31, 8	gt0ap0d 8673	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) # 0 )
33	10, 11, 32	mulap0bbd 8704	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B # 0 )
34		0cnd 8036	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. CC )
35		apsym 8650	. . . 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 8676	. . 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 946	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 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   x. cmul 7901   < clt 8078   <_ cle 8079   -ucneg 8215   # cap 8625

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717

This theorem is referenced by:  prodgt02  8897  prodgt0i  8952  evennn2n  12065