Theorem rexpssxrxp 8002

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 8001	. 2 |- RR C_ RR*
2		xpss12 4734	. 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 3130   X. cxp 4625   RRcr 7810   RR*cxr 7991

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-opab 4066  df-xp 4633  df-xr 7996

This theorem is referenced by:  ltrelxr  8018