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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 3509 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | rrhval.1 | . . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | 2 | eqcomi 2749 | . . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽 |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽) |
5 | fveq2 6922 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅)) | |
6 | rrhval.2 | . . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅) | |
7 | 5, 6 | eqtr4di 2798 | . . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾) |
8 | 4, 7 | oveq12d 7468 | . . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾)) |
9 | fveq2 6922 | . . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅)) | |
10 | 8, 9 | fveq12d 6929 | . . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
11 | df-rrh 33943 | . . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟))) | |
12 | fvex 6935 | . . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V | |
13 | 10, 11, 12 | fvmpt 7031 | . 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 1537 ∈ wcel 2108 Vcvv 3488 ran crn 5701 ‘cfv 6575 (class class class)co 7450 (,)cioo 13409 TopOpenctopn 17483 topGenctg 17499 CnExtccnext 24090 ℚHomcqqh 33920 ℝHomcrrh 33941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6527 df-fun 6577 df-fv 6583 df-ov 7453 df-rrh 33943 |
This theorem is referenced by: rrhcn 33945 rrhqima 33962 rrhre 33969 |
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