![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0tps | Structured version Visualization version GIF version |
Description: The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
Ref | Expression |
---|---|
xrge0tps | ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrstps 23196 | . 2 ⊢ ℝ*𝑠 ∈ TopSp | |
2 | ovex 7456 | . 2 ⊢ (0[,]+∞) ∈ V | |
3 | resstps 23174 | . 2 ⊢ ((ℝ*𝑠 ∈ TopSp ∧ (0[,]+∞) ∈ V) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3461 (class class class)co 7423 0cc0 11154 +∞cpnf 11291 [,]cicc 13376 ↾s cress 17237 ℝ*𝑠cxrs 17510 TopSpctps 22917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fi 9450 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-tset 17280 df-ple 17281 df-ds 17283 df-rest 17432 df-topn 17433 df-topgen 17453 df-ordt 17511 df-xrs 17512 df-ps 18586 df-tsr 18587 df-top 22879 df-topon 22896 df-topsp 22918 df-bases 22932 |
This theorem is referenced by: xrge0tmd 33716 xrge0tmdALT 33717 esumcl 33819 esumgsum 33834 esumf1o 33839 esumss 33861 esumpfinval 33864 esumpfinvalf 33865 esumcocn 33869 sitmcl 34141 |
Copyright terms: Public domain | W3C validator |