Theorem rexpssxrxp 7070

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 7069	. 2 |- RR C_ RR*
2		xpss12 4445	. 2 |- ( ( RR C_ RR* /\ RR C_ RR* ) -> ( RR X. RR ) C_ ( RR* X. RR* ) )
3	1, 1, 2	mp2an 402	1 |- ( RR X. RR ) C_ ( RR* X. RR* )

Syntax hints:   C_ wss 2917   X. cxp 4343   RRcr 6888   RR*cxr 7059

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-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-opab 3819  df-xp 4351  df-xr 7064

This theorem is referenced by:  ltrelxr  7080