Theorem axmulgt0 8126

Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 8024 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
axmulgt0	|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )

Proof of Theorem axmulgt0

Step	Hyp	Ref	Expression
1		ax-pre-mulgt0 8024	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
2		0re 8054	. . . 4 |- 0 e. RR
3		ltxrlt 8120	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
4	2, 3	mpan 424	. . 3 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
5		ltxrlt 8120	. . . 4 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> 0 <RR B ) )
6	2, 5	mpan 424	. . 3 |- ( B e. RR -> ( 0 < B <-> 0 <RR B ) )
7	4, 6	bi2anan9 606	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) <-> ( 0 <RR A /\ 0 <RR B ) ) )
8		remulcl 8035	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
9		ltxrlt 8120	. . 3 |- ( ( 0 e. RR /\ ( A x. B ) e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
10	2, 8, 9	sylancr 414	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
11	1, 7, 10	3imtr4d 203	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2175   class class class wbr 4043  (class class class)co 5934   RRcr 7906   0cc0 7907   <RR cltrr 7911   x. cmul 7912   < clt 8089

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004  ax-mulrcl 8006  ax-rnegex 8016  ax-pre-mulgt0 8024

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4679  df-pnf 8091  df-mnf 8092  df-ltxr 8094

This theorem is referenced by:  mulgt0  8129  mulgt0i  8164  sin02gt0  11994  sinq12gt0  15220