Theorem ltrelsr 7736

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 7728	. 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 4700	. 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 3187	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   C_ wss 3129   <.cop 3595   class class class wbr 4003   {copab 4063   X. cxp 4624  (class class class)co 5874   [cec 6532   +P. cpp 7291   <P cltp 7293   ~R cer 7294   R.cnr 7295   <R cltr 7301

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3135  df-ss 3142  df-opab 4065  df-xp 4632  df-ltr 7728

This theorem is referenced by:  gt0srpr  7746  recexgt0sr  7771  addgt0sr  7773  mulgt0sr  7776  caucvgsrlemcl  7787  caucvgsrlemasr  7788  caucvgsrlemfv  7789  map2psrprg  7803  suplocsrlemb  7804  suplocsrlempr  7805  suplocsrlem  7806  suplocsr  7807  ltresr  7837  axpre-ltirr  7880  axpre-lttrn  7882