Theorem mullt0 8627

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 8407	. . . . 5 |- ( A e. RR -> -u A e. RR )
2	1	adantr 276	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> -u A e. RR )
3		lt0neg1 8615	. . . . 5 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
4	3	biimpa 296	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> 0 < -u A )
5	2, 4	jca 306	. . 3 |- ( ( A e. RR /\ A < 0 ) -> ( -u A e. RR /\ 0 < -u A ) )
6		renegcl 8407	. . . . 5 |- ( B e. RR -> -u B e. RR )
7	6	adantr 276	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> -u B e. RR )
8		lt0neg1 8615	. . . . 5 |- ( B e. RR -> ( B < 0 <-> 0 < -u B ) )
9	8	biimpa 296	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> 0 < -u B )
10	7, 9	jca 306	. . 3 |- ( ( B e. RR /\ B < 0 ) -> ( -u B e. RR /\ 0 < -u B ) )
11		mulgt0 8221	. . 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 289	. 2 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( -u A x. -u B ) )
13		recn 8132	. . . 4 |- ( A e. RR -> A e. CC )
14		recn 8132	. . . 4 |- ( B e. RR -> B e. CC )
15		mul2neg 8544	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) )
16	13, 14, 15	syl2an 289	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -u A x. -u B ) = ( A x. B ) )
17	16	ad2ant2r 509	. 2 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> ( -u A x. -u B ) = ( A x. B ) )
18	12, 17	breqtrd 4109	1 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   x. cmul 8004   < clt 8181   -ucneg 8318

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-sub 8319  df-neg 8320

This theorem is referenced by:  inelr  8731  apsqgt0  8748