Theorem ltrelsr 8095

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 8087	. 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 4844	. 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 3280	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   <.cop 3708   class class class wbr 4125   {copab 4186   X. cxp 4767  (class class class)co 6075   [cec 6795   +P. cpp 7650   <P cltp 7652   ~R cer 7653   R.cnr 7654   <R cltr 7660

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-opab 4188  df-xp 4775  df-ltr 8087

This theorem is referenced by:  gt0srpr  8105  recexgt0sr  8130  addgt0sr  8132  mulgt0sr  8135  caucvgsrlemcl  8146  caucvgsrlemasr  8147  caucvgsrlemfv  8148  map2psrprg  8162  suplocsrlemb  8163  suplocsrlempr  8164  suplocsrlem  8165  suplocsr  8166  ltresr  8196  axpre-ltirr  8239  axpre-lttrn  8241