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 16769 | . 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 7464 | . . 3 ⊢ {〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∈ V | |
3 | tpex 7464 | . . 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 7463 | . 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 2909 | 1 ⊢ ℝ*𝑠 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 ifcif 4466 {ctp 4564 〈cop 4566 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ℝ*cxr 10668 ≤ cle 10670 -𝑒cxne 12498 +𝑒 cxad 12499 ·e cxmu 12500 ndxcnx 16474 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 TopSetcts 16565 lecple 16566 distcds 16568 ordTopcordt 16766 ℝ*𝑠cxrs 16767 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-dif 3938 df-un 3940 df-nul 4291 df-sn 4561 df-pr 4563 df-tp 4565 df-uni 4832 df-xrs 16769 |
This theorem is referenced by: imasdsf1olem 22977 xrslt 30658 xrsmulgzz 30660 xrstos 30661 xrsp0 30663 xrsp1 30664 pnfinf 30807 xrnarchi 30808 |
Copyright terms: Public domain | W3C validator |