Theorem ltrelsr 7957

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 7949	. 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 4800	. 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 3259	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   C_ wss 3200   <.cop 3672   class class class wbr 4088   {copab 4149   X. cxp 4723  (class class class)co 6017   [cec 6699   +P. cpp 7512   <P cltp 7514   ~R cer 7515   R.cnr 7516   <R cltr 7522

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-opab 4151  df-xp 4731  df-ltr 7949

This theorem is referenced by:  gt0srpr  7967  recexgt0sr  7992  addgt0sr  7994  mulgt0sr  7997  caucvgsrlemcl  8008  caucvgsrlemasr  8009  caucvgsrlemfv  8010  map2psrprg  8024  suplocsrlemb  8025  suplocsrlempr  8026  suplocsrlem  8027  suplocsr  8028  ltresr  8058  axpre-ltirr  8101  axpre-lttrn  8103