Theorem rrhval 34711

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 3475	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		rrhval.1	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
3	2	eqcomi 2772	. . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽
4	3	a1i 11	. . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽)
5		fveq2 6911	. . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅))
6		rrhval.2	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅)
7	5, 6	eqtr4di 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾)
8	4, 7	oveq12d 7464	. . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾))
9		fveq2 6911	. . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅))
10	8, 9	fveq12d 6918	. . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		df-rrh 34710	. . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
12		fvex 6924	. . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V
13	10, 11, 12	fvmpt 7019	. 2 ⊢ (𝑅 ∈ V → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
14	1, 13	syl 18	1 ⊢ (𝑅 ∈ 𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ran crn 5657  ‘cfv 6561  (class class class)co 7446  (,)cioo 13506  TopOpenctopn 17624  topGenctg 17640  CnExtccnext 24411  ℚHomcqqh 34685  ℝHomcrrh 34708

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-rrh 34710

This theorem is used by:  rrhcn  34712  rrhqima  34729  rrhre  34736