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Theorem mullt0 9963
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
StepHypRef Expression
1 renegcl 9776 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
21adantr 465 . . . 4  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  -u A  e.  RR )
3 lt0neg1 9949 . . . . 5  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
43biimpa 484 . . . 4  |-  ( ( A  e.  RR  /\  A  <  0 )  -> 
0  <  -u A )
52, 4jca 532 . . 3  |-  ( ( A  e.  RR  /\  A  <  0 )  -> 
( -u A  e.  RR  /\  0  <  -u A
) )
6 renegcl 9776 . . . . 5  |-  ( B  e.  RR  ->  -u B  e.  RR )
76adantr 465 . . . 4  |-  ( ( B  e.  RR  /\  B  <  0 )  ->  -u B  e.  RR )
8 lt0neg1 9949 . . . . 5  |-  ( B  e.  RR  ->  ( B  <  0  <->  0  <  -u B ) )
98biimpa 484 . . . 4  |-  ( ( B  e.  RR  /\  B  <  0 )  -> 
0  <  -u B )
107, 9jca 532 . . 3  |-  ( ( B  e.  RR  /\  B  <  0 )  -> 
( -u B  e.  RR  /\  0  <  -u B
) )
11 mulgt0 9556 . . 3  |-  ( ( ( -u A  e.  RR  /\  0  <  -u A )  /\  ( -u B  e.  RR  /\  0  <  -u B ) )  ->  0  <  ( -u A  x.  -u B
) )
125, 10, 11syl2an 477 . 2  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
0  <  ( -u A  x.  -u B ) )
13 recn 9476 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
14 recn 9476 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
15 mul2neg 9888 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  -u B )  =  ( A  x.  B ) )
1613, 14, 15syl2an 477 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  x.  -u B )  =  ( A  x.  B ) )
1716ad2ant2r 746 . 2  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
( -u A  x.  -u B
)  =  ( A  x.  B ) )
1812, 17breqtrd 4417 1  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
0  <  ( A  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4393  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386    x. cmul 9391    < clt 9522   -ucneg 9700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702
This theorem is referenced by:  msqgt0  9964
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