Theorem ltrelsr 8069

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 8061	. 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 4829	. 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 3274	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   C_ wss 3214   <.cop 3697   class class class wbr 4114   {copab 4175   X. cxp 4752  (class class class)co 6058   [cec 6778   +P. cpp 7624   <P cltp 7626   ~R cer 7627   R.cnr 7628   <R cltr 7634

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 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3220  df-ss 3227  df-opab 4177  df-xp 4760  df-ltr 8061

This theorem is referenced by:  gt0srpr  8079  recexgt0sr  8104  addgt0sr  8106  mulgt0sr  8109  caucvgsrlemcl  8120  caucvgsrlemasr  8121  caucvgsrlemfv  8122  map2psrprg  8136  suplocsrlemb  8137  suplocsrlempr  8138  suplocsrlem  8139  suplocsr  8140  ltresr  8170  axpre-ltirr  8213  axpre-lttrn  8215