Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-rrh Structured version   Visualization version   GIF version

Definition df-rrh 30167
 Description: Define the canonical homomorphism from the real numbers to any complete field, as the extension by continuity of the canonical homomorphism from the rational numbers. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
Assertion
Ref Expression
df-rrh ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))

Detailed syntax breakdown of Definition df-rrh
StepHypRef Expression
1 crrh 30165 . 2 class ℝHom
2 vr . . 3 setvar 𝑟
3 cvv 3231 . . 3 class V
42cv 1522 . . . . 5 class 𝑟
5 cqqh 30144 . . . . 5 class ℚHom
64, 5cfv 5926 . . . 4 class (ℚHom‘𝑟)
7 cioo 12213 . . . . . . 7 class (,)
87crn 5144 . . . . . 6 class ran (,)
9 ctg 16145 . . . . . 6 class topGen
108, 9cfv 5926 . . . . 5 class (topGen‘ran (,))
11 ctopn 16129 . . . . . 6 class TopOpen
124, 11cfv 5926 . . . . 5 class (TopOpen‘𝑟)
13 ccnext 21910 . . . . 5 class CnExt
1410, 12, 13co 6690 . . . 4 class ((topGen‘ran (,))CnExt(TopOpen‘𝑟))
156, 14cfv 5926 . . 3 class (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟))
162, 3, 15cmpt 4762 . 2 class (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
171, 16wceq 1523 1 wff ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
 Colors of variables: wff setvar class This definition is referenced by:  rrhval  30168
 Copyright terms: Public domain W3C validator