Theorem rrhval 34153

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 3461	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		rrhval.1	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
3	2	eqcomi 2745	. . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽
4	3	a1i 11	. . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽)
5		fveq2 6834	. . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅))
6		rrhval.2	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅)
7	5, 6	eqtr4di 2789	. . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾)
8	4, 7	oveq12d 7376	. . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾))
9		fveq2 6834	. . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅))
10	8, 9	fveq12d 6841	. . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		df-rrh 34152	. . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
12		fvex 6847	. . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V
13	10, 11, 12	fvmpt 6941	. 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 1541   ∈ wcel 2113  Vcvv 3440  ran crn 5625  ‘cfv 6492  (class class class)co 7358  (,)cioo 13261  TopOpenctopn 17341  topGenctg 17357  CnExtccnext 24003  ℚHomcqqh 34127  ℝHomcrrh 34150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-rrh 34152

This theorem is referenced by:  rrhcn  34154  rrhqima  34171  rrhre  34178