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Axiom ax-pre-mulgt0 7058
Description: The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 7018. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-mulgt0  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )

Detailed syntax breakdown of Axiom ax-pre-mulgt0
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 6945 . . . 4  class  RR
31, 2wcel 1409 . . 3  wff  A  e.  RR
4 cB . . . 4  class  B
54, 2wcel 1409 . . 3  wff  B  e.  RR
63, 5wa 101 . 2  wff  ( A  e.  RR  /\  B  e.  RR )
7 cc0 6946 . . . . 5  class  0
8 cltrr 6950 . . . . 5  class  <RR
97, 1, 8wbr 3791 . . . 4  wff  0  <RR  A
107, 4, 8wbr 3791 . . . 4  wff  0  <RR  B
119, 10wa 101 . . 3  wff  ( 0 
<RR  A  /\  0  <RR  B )
12 cmul 6951 . . . . 5  class  x.
131, 4, 12co 5539 . . . 4  class  ( A  x.  B )
147, 13, 8wbr 3791 . . 3  wff  0  <RR  ( A  x.  B
)
1511, 14wi 4 . 2  wff  ( ( 0  <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B
) )
166, 15wi 4 1  wff  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
Colors of variables: wff set class
This axiom is referenced by:  axmulgt0  7149
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