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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 15985 | . 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 6855 | . . 3 ⊢ {〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∈ V | |
3 | tpex 6855 | . . 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 6854 | . 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 2684 | 1 ⊢ ℝ*𝑠 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ifcif 4036 {ctp 4129 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝ*cxr 9952 ≤ cle 9954 -𝑒cxne 11819 +𝑒 cxad 11820 ·e cxmu 11821 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 TopSetcts 15774 lecple 15775 distcds 15777 ordTopcordt 15982 ℝ*𝑠cxrs 15983 |
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-8 1979 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 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 df-tp 4130 df-uni 4373 df-xrs 15985 |
This theorem is referenced by: imasdsf1olem 21988 xrslt 29007 xrsmulgzz 29009 xrstos 29010 xrsp0 29012 xrsp1 29013 pnfinf 29068 xrnarchi 29069 |
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