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Mirrors > Home > MPE Home > Th. List > xrsex | Structured version Visualization version GIF version |
Description: The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsex | ⊢ ℝ*𝑠 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xrs 17448 | . 2 ⊢ ℝ*𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩}) | |
2 | tpex 7734 | . . 3 ⊢ {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∈ V | |
3 | tpex 7734 | . . 3 ⊢ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩} ∈ V | |
4 | 2, 3 | unex 7733 | . 2 ⊢ ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩}) ∈ V |
5 | 1, 4 | eqeltri 2830 | 1 ⊢ ℝ*𝑠 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ∪ cun 3947 ifcif 4529 {ctp 4633 ⟨cop 4635 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 ℝ*cxr 11247 ≤ cle 11249 -𝑒cxne 13089 +𝑒 cxad 13090 ·e cxmu 13091 ndxcnx 17126 Basecbs 17144 +gcplusg 17197 .rcmulr 17198 TopSetcts 17203 lecple 17204 distcds 17206 ordTopcordt 17445 ℝ*𝑠cxrs 17446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-sn 4630 df-pr 4632 df-tp 4634 df-uni 4910 df-xrs 17448 |
This theorem is referenced by: imasdsf1olem 23879 xrslt 32177 xrsmulgzz 32179 xrstos 32180 xrsp0 32182 xrsp1 32183 pnfinf 32329 xrnarchi 32330 |
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