Theorem ltrelsr 7886

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 7878	. 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 4767	. 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 3233	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2178   C_ wss 3174   <.cop 3646   class class class wbr 4059   {copab 4120   X. cxp 4691  (class class class)co 5967   [cec 6641   +P. cpp 7441   <P cltp 7443   ~R cer 7444   R.cnr 7445   <R cltr 7451

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187  df-opab 4122  df-xp 4699  df-ltr 7878

This theorem is referenced by:  gt0srpr  7896  recexgt0sr  7921  addgt0sr  7923  mulgt0sr  7926  caucvgsrlemcl  7937  caucvgsrlemasr  7938  caucvgsrlemfv  7939  map2psrprg  7953  suplocsrlemb  7954  suplocsrlempr  7955  suplocsrlem  7956  suplocsr  7957  ltresr  7987  axpre-ltirr  8030  axpre-lttrn  8032