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 3514 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | rrhval.1 | . . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | 2 | eqcomi 2832 | . . . . . 6 ⊢ (topGen‘ran (,)) = 𝐽 |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝑟 = 𝑅 → (topGen‘ran (,)) = 𝐽) |
5 | fveq2 6672 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = (TopOpen‘𝑅)) | |
6 | rrhval.2 | . . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑅) | |
7 | 5, 6 | syl6eqr 2876 | . . . . 5 ⊢ (𝑟 = 𝑅 → (TopOpen‘𝑟) = 𝐾) |
8 | 4, 7 | oveq12d 7176 | . . . 4 ⊢ (𝑟 = 𝑅 → ((topGen‘ran (,))CnExt(TopOpen‘𝑟)) = (𝐽CnExt𝐾)) |
9 | fveq2 6672 | . . . 4 ⊢ (𝑟 = 𝑅 → (ℚHom‘𝑟) = (ℚHom‘𝑅)) | |
10 | 8, 9 | fveq12d 6679 | . . 3 ⊢ (𝑟 = 𝑅 → (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
11 | df-rrh 31238 | . . 3 ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟))) | |
12 | fvex 6685 | . . 3 ⊢ ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ V | |
13 | 10, 11, 12 | fvmpt 6770 | . 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 2114 Vcvv 3496 ran crn 5558 ‘cfv 6357 (class class class)co 7158 (,)cioo 12741 TopOpenctopn 16697 topGenctg 16713 CnExtccnext 22669 ℚHomcqqh 31215 ℝHomcrrh 31236 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-rrh 31238 |
This theorem is referenced by: rrhcn 31240 rrhqima 31257 rrhre 31264 |
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