Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexpssxrxp GIF version

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 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)

Proof of Theorem rexpssxrxp
StepHypRef Expression
1 ressxr 7069 . 2 ℝ ⊆ ℝ*
2 xpss12 4445 . 2 ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
31, 1, 2mp2an 402 1 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
 Colors of variables: wff set class Syntax hints:   ⊆ wss 2917   × cxp 4343  ℝcr 6888  ℝ*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
 Copyright terms: Public domain W3C validator