Theorem ltrelsr 6915

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 6907	. 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 4432	. 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 3029	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   C_ wss 2973   <.cop 3401   class class class wbr 3785   {copab 3838   X. cxp 4361  (class class class)co 5532   [cec 6127   +P. cpp 6483   <P cltp 6485   ~R cer 6486   R.cnr 6487   <R cltr 6493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-opab 3840  df-xp 4369  df-ltr 6907

This theorem is referenced by:  gt0srpr  6925  recexgt0sr  6950  addgt0sr  6952  mulgt0sr  6954  caucvgsrlemcl  6965  caucvgsrlemasr  6966  caucvgsrlemfv  6967  ltresr  7007  axpre-ltirr  7048  axpre-lttrn  7050