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Axiom ax-pre-mulext 7059
Description: Strong extensionality of multiplication (expressed in terms of  <RR). Axiom for real and complex numbers, justified by theorem axpre-mulext 7019

(Contributed by Jim Kingdon, 18-Feb-2020.)

Assertion
Ref Expression
ax-pre-mulext  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <RR  ( B  x.  C )  ->  ( A  <RR  B  \/  B  <RR  A ) ) )

Detailed syntax breakdown of Axiom ax-pre-mulext
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
6 cC . . . 4  class  C
76, 2wcel 1409 . . 3  wff  C  e.  RR
83, 5, 7w3a 896 . 2  wff  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
9 cmul 6951 . . . . 5  class  x.
101, 6, 9co 5539 . . . 4  class  ( A  x.  C )
114, 6, 9co 5539 . . . 4  class  ( B  x.  C )
12 cltrr 6950 . . . 4  class  <RR
1310, 11, 12wbr 3791 . . 3  wff  ( A  x.  C )  <RR  ( B  x.  C )
141, 4, 12wbr 3791 . . . 4  wff  A  <RR  B
154, 1, 12wbr 3791 . . . 4  wff  B  <RR  A
1614, 15wo 639 . . 3  wff  ( A 
<RR  B  \/  B  <RR  A )
1713, 16wi 4 . 2  wff  ( ( A  x.  C ) 
<RR  ( B  x.  C
)  ->  ( A  <RR  B  \/  B  <RR  A ) )
188, 17wi 4 1  wff  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <RR  ( B  x.  C )  ->  ( A  <RR  B  \/  B  <RR  A ) ) )
Colors of variables: wff set class
This axiom is referenced by:  remulext1  7663
  Copyright terms: Public domain W3C validator