Theorem mullt0 8507

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 8287	. . . . 5 |- ( A e. RR -> -u A e. RR )
2	1	adantr 276	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> -u A e. RR )
3		lt0neg1 8495	. . . . 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 8287	. . . . 5 |- ( B e. RR -> -u B e. RR )
7	6	adantr 276	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> -u B e. RR )
8		lt0neg1 8495	. . . . 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 8101	. . 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 8012	. . . 4 |- ( A e. RR -> A e. CC )
14		recn 8012	. . . 4 |- ( B e. RR -> B e. CC )
15		mul2neg 8424	. . . 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 4059	1 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   class class class wbr 4033  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   x. cmul 7884   < clt 8061   -ucneg 8198

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltadd 7995  ax-pre-mulgt0 7996

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-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-sub 8199  df-neg 8200

This theorem is referenced by:  inelr  8611  apsqgt0  8628