Theorem ltrelsr 7853

Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)

Assertion

Ref	Expression
ltrelsr	|- <R C_ ( R. X. R. )

Proof of Theorem ltrelsr

Dummy variables x y z w v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltr 7845	. 2 |- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) }
2		opabssxp 4750	. 2 |- { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) } C_ ( R. X. R. )
3	1, 2	eqsstri 3225	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1515   e. wcel 2176   C_ wss 3166   <.cop 3636   class class class wbr 4045   {copab 4105   X. cxp 4674  (class class class)co 5946   [cec 6620   +P. cpp 7408   <P cltp 7410   ~R cer 7411   R.cnr 7412   <R cltr 7418

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-opab 4107  df-xp 4682  df-ltr 7845

This theorem is referenced by:  gt0srpr  7863  recexgt0sr  7888  addgt0sr  7890  mulgt0sr  7893  caucvgsrlemcl  7904  caucvgsrlemasr  7905  caucvgsrlemfv  7906  map2psrprg  7920  suplocsrlemb  7921  suplocsrlempr  7922  suplocsrlem  7923  suplocsr  7924  ltresr  7954  axpre-ltirr  7997  axpre-lttrn  7999