Theorem ltrelsr 7700

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 7692	. 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 4685	. 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 3179	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   C_ wss 3121   <.cop 3586   class class class wbr 3989   {copab 4049   X. cxp 4609  (class class class)co 5853   [cec 6511   +P. cpp 7255   <P cltp 7257   ~R cer 7258   R.cnr 7259   <R cltr 7265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-opab 4051  df-xp 4617  df-ltr 7692

This theorem is referenced by:  gt0srpr  7710  recexgt0sr  7735  addgt0sr  7737  mulgt0sr  7740  caucvgsrlemcl  7751  caucvgsrlemasr  7752  caucvgsrlemfv  7753  map2psrprg  7767  suplocsrlemb  7768  suplocsrlempr  7769  suplocsrlem  7770  suplocsr  7771  ltresr  7801  axpre-ltirr  7844  axpre-lttrn  7846