Theorem ltrelsr 7805

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 7797	. 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 4737	. 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 3215	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   C_ wss 3157   <.cop 3625   class class class wbr 4033   {copab 4093   X. cxp 4661  (class class class)co 5922   [cec 6590   +P. cpp 7360   <P cltp 7362   ~R cer 7363   R.cnr 7364   <R cltr 7370

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-opab 4095  df-xp 4669  df-ltr 7797

This theorem is referenced by:  gt0srpr  7815  recexgt0sr  7840  addgt0sr  7842  mulgt0sr  7845  caucvgsrlemcl  7856  caucvgsrlemasr  7857  caucvgsrlemfv  7858  map2psrprg  7872  suplocsrlemb  7873  suplocsrlempr  7874  suplocsrlem  7875  suplocsr  7876  ltresr  7906  axpre-ltirr  7949  axpre-lttrn  7951