Theorem rrhval 31946

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	⊢ (𝑅 ∈ 𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))

Proof of Theorem rrhval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		rrhval.1	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
3	2	eqcomi 2747	. . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽
4	3	a1i 11	. . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽)
5		fveq2 6774	. . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅))
6		rrhval.2	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅)
7	5, 6	eqtr4di 2796	. . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾)
8	4, 7	oveq12d 7293	. . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾))
9		fveq2 6774	. . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅))
10	8, 9	fveq12d 6781	. . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		df-rrh 31945	. . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
12		fvex 6787	. . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V
13	10, 11, 12	fvmpt 6875	. 2 ⊢ (𝑅 ∈ V → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
14	1, 13	syl 17	1 ⊢ (𝑅 ∈ 𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ran crn 5590  ‘cfv 6433  (class class class)co 7275  (,)cioo 13079  TopOpenctopn 17132  topGenctg 17148  CnExtccnext 23210  ℚHomcqqh 31922  ℝHomcrrh 31943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-rrh 31945

This theorem is referenced by:  rrhcn  31947  rrhqima  31964  rrhre  31971