Theorem mullt0 8242

Description: The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)

Assertion

Ref	Expression
mullt0	|- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )

Proof of Theorem mullt0

Step	Hyp	Ref	Expression
1		renegcl 8023	. . . . 5 |- ( A e. RR -> -u A e. RR )
2	1	adantr 274	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> -u A e. RR )
3		lt0neg1 8230	. . . . 5 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
4	3	biimpa 294	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> 0 < -u A )
5	2, 4	jca 304	. . 3 |- ( ( A e. RR /\ A < 0 ) -> ( -u A e. RR /\ 0 < -u A ) )
6		renegcl 8023	. . . . 5 |- ( B e. RR -> -u B e. RR )
7	6	adantr 274	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> -u B e. RR )
8		lt0neg1 8230	. . . . 5 |- ( B e. RR -> ( B < 0 <-> 0 < -u B ) )
9	8	biimpa 294	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> 0 < -u B )
10	7, 9	jca 304	. . 3 |- ( ( B e. RR /\ B < 0 ) -> ( -u B e. RR /\ 0 < -u B ) )
11		mulgt0 7839	. . 3 |- ( ( ( -u A e. RR /\ 0 < -u A ) /\ ( -u B e. RR /\ 0 < -u B ) ) -> 0 < ( -u A x. -u B ) )
12	5, 10, 11	syl2an 287	. 2 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( -u A x. -u B ) )
13		recn 7753	. . . 4 |- ( A e. RR -> A e. CC )
14		recn 7753	. . . 4 |- ( B e. RR -> B e. CC )
15		mul2neg 8160	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) )
16	13, 14, 15	syl2an 287	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -u A x. -u B ) = ( A x. B ) )
17	16	ad2ant2r 500	. 2 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> ( -u A x. -u B ) = ( A x. B ) )
18	12, 17	breqtrd 3954	1 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   x. cmul 7625   < clt 7800   -ucneg 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltadd 7736  ax-pre-mulgt0 7737

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-ltxr 7805  df-sub 7935  df-neg 7936

This theorem is referenced by:  inelr  8346  apsqgt0  8363