Theorem mullt0 8399

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 8180	. . . . 5 |- ( A e. RR -> -u A e. RR )
2	1	adantr 274	. . . 4 |- ( ( A e. RR /\ A < 0 ) -> -u A e. RR )
3		lt0neg1 8387	. . . . 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 8180	. . . . 5 |- ( B e. RR -> -u B e. RR )
7	6	adantr 274	. . . 4 |- ( ( B e. RR /\ B < 0 ) -> -u B e. RR )
8		lt0neg1 8387	. . . . 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 7994	. . 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 7907	. . . 4 |- ( A e. RR -> A e. CC )
14		recn 7907	. . . 4 |- ( B e. RR -> B e. CC )
15		mul2neg 8317	. . . 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 506	. 2 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> ( -u A x. -u B ) = ( A x. B ) )
18	12, 17	breqtrd 4015	1 |- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   x. cmul 7779   < clt 7954   -ucneg 8091

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltadd 7890  ax-pre-mulgt0 7891

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093

This theorem is referenced by:  inelr  8503  apsqgt0  8520