Theorem ltrelsr 7284

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 7276	. 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 4512	. 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 3056	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 102   = wceq 1289   E.wex 1426   e. wcel 1438   C_ wss 2999   <.cop 3449   class class class wbr 3845   {copab 3898   X. cxp 4436  (class class class)co 5652   [cec 6290   +P. cpp 6852   <P cltp 6854   ~R cer 6855   R.cnr 6856   <R cltr 6862

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3005  df-ss 3012  df-opab 3900  df-xp 4444  df-ltr 7276

This theorem is referenced by:  gt0srpr  7294  recexgt0sr  7319  addgt0sr  7321  mulgt0sr  7323  caucvgsrlemcl  7334  caucvgsrlemasr  7335  caucvgsrlemfv  7336  ltresr  7376  axpre-ltirr  7417  axpre-lttrn  7419