Theorem ltrelsr 8105

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 8097	. 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 4849	. 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 3280	1 |- <R C_ ( R. X. R. )

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   <.cop 3712   class class class wbr 4130   {copab 4191   X. cxp 4772  (class class class)co 6085   [cec 6805   +P. cpp 7660   <P cltp 7662   ~R cer 7663   R.cnr 7664   <R cltr 7670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-opab 4193  df-xp 4780  df-ltr 8097

This theorem is used by:  gt0srpr  8115  recexgt0sr  8140  addgt0sr  8142  mulgt0sr  8145  caucvgsrlemcl  8156  caucvgsrlemasr  8157  caucvgsrlemfv  8158  map2psrprg  8172  suplocsrlemb  8173  suplocsrlempr  8174  suplocsrlem  8175  suplocsr  8176  ltresr  8206  axpre-ltirr  8249  axpre-lttrn  8251