Theorem rexpssxrxp 8099

Description: The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Assertion

Ref	Expression
rexpssxrxp	|- ( RR X. RR ) C_ ( RR* X. RR* )

Proof of Theorem rexpssxrxp

Step	Hyp	Ref	Expression
1		ressxr 8098	. 2 |- RR C_ RR*
2		xpss12 4780	. 2 |- ( ( RR C_ RR* /\ RR C_ RR* ) -> ( RR X. RR ) C_ ( RR* X. RR* ) )
3	1, 1, 2	mp2an 426	1 |- ( RR X. RR ) C_ ( RR* X. RR* )

Syntax hints:   C_ wss 3165   X. cxp 4671   RRcr 7906   RR*cxr 8088

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-opab 4105  df-xp 4679  df-xr 8093

This theorem is referenced by:  ltrelxr  8115