Theorem rexpssxrxp 7803

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 7802	. 2 |- RR C_ RR*
2		xpss12 4641	. 2 |- ( ( RR C_ RR* /\ RR C_ RR* ) -> ( RR X. RR ) C_ ( RR* X. RR* ) )
3	1, 1, 2	mp2an 422	1 |- ( RR X. RR ) C_ ( RR* X. RR* )

Syntax hints:   C_ wss 3066   X. cxp 4532   RRcr 7612   RR*cxr 7792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-opab 3985  df-xp 4540  df-xr 7797

This theorem is referenced by:  ltrelxr  7818