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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrexthaus | Structured version Visualization version GIF version |
Description: The topology of an extension of ℝ is Hausdorff. (Contributed by Thierry Arnoux, 7-Sep-2018.) |
Ref | Expression |
---|---|
rrexthaus.1 | ⊢ 𝐾 = (TopOpen‘𝑅) |
Ref | Expression |
---|---|
rrexthaus | ⊢ (𝑅 ∈ ℝExt → 𝐾 ∈ Haus) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrextnrg 30892 | . . . 4 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing) | |
2 | nrgngp 22974 | . . . 4 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | |
3 | ngpxms 22913 | . . . 4 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ ∞MetSp) |
5 | rrexthaus.1 | . . . 4 ⊢ 𝐾 = (TopOpen‘𝑅) | |
6 | eqid 2778 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2778 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
8 | 5, 6, 7 | xmstopn 22764 | . . 3 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))) |
9 | 4, 8 | syl 17 | . 2 ⊢ (𝑅 ∈ ℝExt → 𝐾 = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))) |
10 | 6, 7 | xmsxmet 22769 | . . 3 ⊢ (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅))) |
11 | eqid 2778 | . . . 4 ⊢ (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) | |
12 | 11 | methaus 22833 | . . 3 ⊢ (((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)) → (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) ∈ Haus) |
13 | 4, 10, 12 | 3syl 18 | . 2 ⊢ (𝑅 ∈ ℝExt → (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) ∈ Haus) |
14 | 9, 13 | eqeltrd 2866 | 1 ⊢ (𝑅 ∈ ℝExt → 𝐾 ∈ Haus) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 × cxp 5405 ↾ cres 5409 ‘cfv 6188 Basecbs 16339 distcds 16430 TopOpenctopn 16551 ∞Metcxmet 20232 MetOpencmopn 20237 Hauscha 21620 ∞MetSpcxms 22630 NrmGrpcngp 22890 NrmRingcnrg 22892 ℝExt crrext 30885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-icc 12561 df-topgen 16573 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-top 21206 df-topon 21223 df-topsp 21245 df-bases 21258 df-haus 21627 df-xms 22633 df-ms 22634 df-ngp 22896 df-nrg 22898 df-rrext 30890 |
This theorem is referenced by: rrhqima 30905 |
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