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Axiom ax-pre-mulgt0 8814
Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 8790. Normally new proofs would use axmulgt0 8897. (New usage is discouraged.) (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 8736 . . . 4  class  RR
31, 2wcel 1684 . . 3  wff  A  e.  RR
4 cB . . . 4  class  B
54, 2wcel 1684 . . 3  wff  B  e.  RR
63, 5wa 358 . 2  wff  ( A  e.  RR  /\  B  e.  RR )
7 cc0 8737 . . . . 5  class  0
8 cltrr 8741 . . . . 5  class  <RR
97, 1, 8wbr 4023 . . . 4  wff  0  <RR  A
107, 4, 8wbr 4023 . . . 4  wff  0  <RR  B
119, 10wa 358 . . 3  wff  ( 0 
<RR  A  /\  0  <RR  B )
12 cmul 8742 . . . . 5  class  x.
131, 4, 12co 5858 . . . 4  class  ( A  x.  B )
147, 13, 8wbr 4023 . . 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  8897
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