Theorem rrhval 34514

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 3474	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		rrhval.1	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
3	2	eqcomi 2771	. . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽
4	3	a1i 11	. . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽)
5		fveq2 6882	. . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅))
6		rrhval.2	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅)
7	5, 6	eqtr4di 2815	. . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾)
8	4, 7	oveq12d 7435	. . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾))
9		fveq2 6882	. . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅))
10	8, 9	fveq12d 6889	. . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		df-rrh 34513	. . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
12		fvex 6895	. . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V
13	10, 11, 12	fvmpt 6990	. 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 2145  Vcvv 3453  ran crn 5660  ‘cfv 6537  (class class class)co 7417  (,)cioo 13402  TopOpenctopn 17512  topGenctg 17528  CnExtccnext 24291  ℚHomcqqh 34488  ℝHomcrrh 34511

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-rrh 34513

This theorem is used by:  rrhcn  34515  rrhqima  34532  rrhre  34539