Theorem ltrelsr 6823

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 6815	. 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 4414	. 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 2975	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   <.cop 3378   class class class wbr 3764   {copab 3817   X. cxp 4343  (class class class)co 5512   [cec 6104   +P. cpp 6391   <P cltp 6393   ~R cer 6394   R.cnr 6395   <R cltr 6401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931  df-opab 3819  df-xp 4351  df-ltr 6815

This theorem is referenced by:  gt0srpr  6833  recexgt0sr  6858  addgt0sr  6860  mulgt0sr  6862  caucvgsrlemcl  6873  caucvgsrlemasr  6874  caucvgsrlemfv  6875  ltresr  6915  axpre-ltirr  6956  axpre-lttrn  6958