Theorem prodgt0 9143

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 536	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> B e. RR )
2	1	renegcld 8670	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u B e. RR )
3		simplll 535	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> A e. RR )
4	3	renegcld 8670	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> -u A e. RR )
5		simplr 529	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. RR )
6	5	lt0neg1d 8806	. . . . . . 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 533	. . . . . . . . 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 8318	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> A e. CC )
11	5	recnd 8318	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B e. CC )
12	10, 11	mul2negd 8703	. . . . . . . . 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 4142	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < ( -u A x. -u B ) )
14	10	negcld 8587	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u A e. CC )
15	11	negcld 8587	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -u B e. CC )
16	14, 15	mulcomd 8311	. . . . . . . 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 4140	. . . . . . 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 9142	. . . . . 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 1275	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 < -u A )
21	3	lt0neg1d 8806	. . . . 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 537	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 <_ A )
24		0red 8291	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) /\ B < 0 ) -> 0 e. RR )
25	24, 3	lenltd 8407	. . . . 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 667	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> -. B < 0 )
28		0red 8291	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. RR )
29	28, 5	lenltd 8407	. . 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 8320	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. RR )
32	31, 8	gt0ap0d 8920	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) # 0 )
33	10, 11, 32	mulap0bbd 8951	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> B # 0 )
34		0cnd 8283	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 e. CC )
35		apsym 8897	. . . 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 8923	. . 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 953	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 2205   class class class wbr 4114  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   x. cmul 8148   < clt 8324   <_ cle 8325   -ucneg 8461   # cap 8872

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964

This theorem is referenced by:  prodgt02  9144  prodgt0i  9199  evennn2n  12594