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Theorem rrhval 33934
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 3504 . 2 (𝑅𝑉𝑅 ∈ V)
2 rrhval.1 . . . . . . 7 𝐽 = (topGen‘ran (,))
32eqcomi 2743 . . . . . 6 (topGen‘ran (,)) = 𝐽
43a1i 11 . . . . 5 (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽)
5 fveq2 6919 . . . . . 6 (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅))
6 rrhval.2 . . . . . 6 𝐾 = (TopOpen‘𝑅)
75, 6eqtr4di 2792 . . . . 5 (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾)
84, 7oveq12d 7463 . . . 4 (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾))
9 fveq2 6919 . . . 4 (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅))
108, 9fveq12d 6926 . . 3 (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11 df-rrh 33933 . . 3 ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
12 fvex 6932 . . 3 ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V
1310, 11, 12fvmpt 7027 . 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 2103  Vcvv 3482  ran crn 5700  cfv 6572  (class class class)co 7445  (,)cioo 13403  TopOpenctopn 17476  topGenctg 17492  CnExtccnext 24081  ℚHomcqqh 33910  ℝHomcrrh 33931
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-rrh 33933
This theorem is referenced by:  rrhcn  33935  rrhqima  33952  rrhre  33959
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