Theorem ltrelsr 7751

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 7743	. 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 4712	. 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 3199	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1363   E.wex 1502   e. wcel 2158   C_ wss 3141   <.cop 3607   class class class wbr 4015   {copab 4075   X. cxp 4636  (class class class)co 5888   [cec 6547   +P. cpp 7306   <P cltp 7308   ~R cer 7309   R.cnr 7310   <R cltr 7316

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-in 3147  df-ss 3154  df-opab 4077  df-xp 4644  df-ltr 7743

This theorem is referenced by:  gt0srpr  7761  recexgt0sr  7786  addgt0sr  7788  mulgt0sr  7791  caucvgsrlemcl  7802  caucvgsrlemasr  7803  caucvgsrlemfv  7804  map2psrprg  7818  suplocsrlemb  7819  suplocsrlempr  7820  suplocsrlem  7821  suplocsr  7822  ltresr  7852  axpre-ltirr  7895  axpre-lttrn  7897