![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17430 | . 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 7717 | . . 3 ⊢ {〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∈ V | |
3 | tpex 7717 | . . 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 7716 | . 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 2828 | 1 ⊢ ℝ*𝑠 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3473 ∪ cun 3942 ifcif 4522 {ctp 4626 〈cop 4628 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 ℝ*cxr 11229 ≤ cle 11231 -𝑒cxne 13071 +𝑒 cxad 13072 ·e cxmu 13073 ndxcnx 17108 Basecbs 17126 +gcplusg 17179 .rcmulr 17180 TopSetcts 17185 lecple 17186 distcds 17188 ordTopcordt 17427 ℝ*𝑠cxrs 17428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-sn 4623 df-pr 4625 df-tp 4627 df-uni 4902 df-xrs 17430 |
This theorem is referenced by: imasdsf1olem 23808 xrslt 32048 xrsmulgzz 32050 xrstos 32051 xrsp0 32053 xrsp1 32054 pnfinf 32200 xrnarchi 32201 |
Copyright terms: Public domain | W3C validator |