Theorem rrhval 33965

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 3459	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		rrhval.1	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
3	2	eqcomi 2738	. . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽
4	3	a1i 11	. . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽)
5		fveq2 6826	. . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅))
6		rrhval.2	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅)
7	5, 6	eqtr4di 2782	. . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾)
8	4, 7	oveq12d 7371	. . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾))
9		fveq2 6826	. . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅))
10	8, 9	fveq12d 6833	. . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		df-rrh 33964	. . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
12		fvex 6839	. . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V
13	10, 11, 12	fvmpt 6934	. 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 1540   ∈ wcel 2109  Vcvv 3438  ran crn 5624  ‘cfv 6486  (class class class)co 7353  (,)cioo 13266  TopOpenctopn 17343  topGenctg 17359  CnExtccnext 23962  ℚHomcqqh 33939  ℝHomcrrh 33962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-rrh 33964

This theorem is referenced by:  rrhcn  33966  rrhqima  33983  rrhre  33990