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Theorem rrhval 33944
Description: Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.)
Hypotheses
Ref Expression
rrhval.1 𝐽 = (topGen‘ran (,))
rrhval.2 𝐾 = (TopOpen‘𝑅)
Assertion
Ref Expression
rrhval (𝑅𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))

Proof of Theorem rrhval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elex 3509 . 2 (𝑅𝑉𝑅 ∈ V)
2 rrhval.1 . . . . . . 7 𝐽 = (topGen‘ran (,))
32eqcomi 2749 . . . . . 6 (topGen‘ran (,)) = 𝐽
43a1i 11 . . . . 5 (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽)
5 fveq2 6922 . . . . . 6 (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅))
6 rrhval.2 . . . . . 6 𝐾 = (TopOpen‘𝑅)
75, 6eqtr4di 2798 . . . . 5 (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾)
84, 7oveq12d 7468 . . . 4 (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾))
9 fveq2 6922 . . . 4 (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅))
108, 9fveq12d 6929 . . 3 (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11 df-rrh 33943 . . 3 ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
12 fvex 6935 . . 3 ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V
1310, 11, 12fvmpt 7031 . 2 (𝑅 ∈ V → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
141, 13syl 17 1 (𝑅𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  Vcvv 3488  ran crn 5701  cfv 6575  (class class class)co 7450  (,)cioo 13409  TopOpenctopn 17483  topGenctg 17499  CnExtccnext 24090  ℚHomcqqh 33920  ℝHomcrrh 33941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-rrh 33943
This theorem is referenced by:  rrhcn  33945  rrhqima  33962  rrhre  33969
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