Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrhval | Structured version Visualization version GIF version |
Description: Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.) |
Ref | Expression |
---|---|
rrhval.1 | ⊢ 𝐽 = (topGen‘ran (,)) |
rrhval.2 | ⊢ 𝐾 = (TopOpen‘𝑅) |
Ref | Expression |
---|---|
rrhval | ⊢ (𝑅 ∈ 𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | rrhval.1 | . . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | 2 | eqcomi 2619 | . . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽 |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽) |
5 | fveq2 6103 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅)) | |
6 | rrhval.2 | . . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅) | |
7 | 5, 6 | syl6eqr 2662 | . . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾) |
8 | 4, 7 | oveq12d 6567 | . . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾)) |
9 | fveq2 6103 | . . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅)) | |
10 | 8, 9 | fveq12d 6109 | . . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
11 | df-rrh 29367 | . . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟))) | |
12 | fvex 6113 | . . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V | |
13 | 10, 11, 12 | fvmpt 6191 | . 2 ⊢ (𝑅 ∈ V → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
14 | 1, 13 | syl 17 | 1 ⊢ (𝑅 ∈ 𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ran crn 5039 ‘cfv 5804 (class class class)co 6549 (,)cioo 12046 TopOpenctopn 15905 topGenctg 15921 CnExtccnext 21673 ℚHomcqqh 29344 ℝHomcrrh 29365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-rrh 29367 |
This theorem is referenced by: rrhcn 29369 rrhqima 29386 rrhre 29393 |
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