Theorem ltrelsr 8053

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 8045	. 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 4824	. 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 3270	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   C_ wss 3211   <.cop 3692   class class class wbr 4109   {copab 4170   X. cxp 4747  (class class class)co 6050   [cec 6765   +P. cpp 7608   <P cltp 7610   ~R cer 7611   R.cnr 7612   <R cltr 7618

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224  df-opab 4172  df-xp 4755  df-ltr 8045

This theorem is referenced by:  gt0srpr  8063  recexgt0sr  8088  addgt0sr  8090  mulgt0sr  8093  caucvgsrlemcl  8104  caucvgsrlemasr  8105  caucvgsrlemfv  8106  map2psrprg  8120  suplocsrlemb  8121  suplocsrlempr  8122  suplocsrlem  8123  suplocsr  8124  ltresr  8154  axpre-ltirr  8197  axpre-lttrn  8199