Theorem mullt0 7475

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 7272	. . . . 5 |- ( A e. RR -> -u A e. RR )
2	1	adantr 261	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> -u A e. RR )
3		lt0neg1 7463	. . . . 5 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
4	3	biimpa 280	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> 0 < -u A )
5	2, 4	jca 290	. . 3 |- ( ( A e. RR /\ A < 0 ) -> ( -u A e. RR /\ 0 < -u A ) )
6		renegcl 7272	. . . . 5 |- ( B e. RR -> -u B e. RR )
7	6	adantr 261	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> -u B e. RR )
8		lt0neg1 7463	. . . . 5 |- ( B e. RR -> ( B < 0 <-> 0 < -u B ) )
9	8	biimpa 280	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> 0 < -u B )
10	7, 9	jca 290	. . 3 |- ( ( B e. RR /\ B < 0 ) -> ( -u B e. RR /\ 0 < -u B ) )
11		mulgt0 7093	. . 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 273	. 2 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( -u A x. -u B ) )
13		recn 7014	. . . 4 |- ( A e. RR -> A e. CC )
14		recn 7014	. . . 4 |- ( B e. RR -> B e. CC )
15		mul2neg 7395	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) )
16	13, 14, 15	syl2an 273	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -u A x. -u B ) = ( A x. B ) )
17	16	ad2ant2r 478	. 2 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> ( -u A x. -u B ) = ( A x. B ) )
18	12, 17	breqtrd 3788	1 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   class class class wbr 3764  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   x. cmul 6894   < clt 7060   -ucneg 7183

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltadd 7000  ax-pre-mulgt0 7001

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-ltxr 7065  df-sub 7184  df-neg 7185

This theorem is referenced by:  inelr  7575  apsqgt0  7592