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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 17547 | . 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 7766 | . . 3 ⊢ {〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∈ V | |
| 3 | tpex 7766 | . . 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 7764 | . 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 2837 | 1 ⊢ ℝ*𝑠 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ifcif 4525 {ctp 4630 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℝ*cxr 11294 ≤ cle 11296 -𝑒cxne 13151 +𝑒 cxad 13152 ·e cxmu 13153 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 TopSetcts 17303 lecple 17304 distcds 17306 ordTopcordt 17544 ℝ*𝑠cxrs 17545 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-tp 4631 df-uni 4908 df-xrs 17547 |
| This theorem is referenced by: imasdsf1olem 24383 xrslt 33009 xrsmulgzz 33011 xrstos 33012 xrsp0 33014 xrsp1 33015 pnfinf 33190 xrnarchi 33191 |
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