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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 17549 | . 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 7765 | . . 3 ⊢ {〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∈ V | |
3 | tpex 7765 | . . 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 7763 | . 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 2835 | 1 ⊢ ℝ*𝑠 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ifcif 4531 {ctp 4635 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ℝ*cxr 11292 ≤ cle 11294 -𝑒cxne 13149 +𝑒 cxad 13150 ·e cxmu 13151 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 TopSetcts 17304 lecple 17305 distcds 17307 ordTopcordt 17546 ℝ*𝑠cxrs 17547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-tp 4636 df-uni 4913 df-xrs 17549 |
This theorem is referenced by: imasdsf1olem 24399 xrslt 32992 xrsmulgzz 32994 xrstos 32995 xrsp0 32997 xrsp1 32998 pnfinf 33173 xrnarchi 33174 |
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