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Mirrors > Home > MPE Home > Th. List > metreg | Structured version Visualization version GIF version |
Description: A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Ref | Expression |
---|---|
metnrm.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
metreg | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metnrm.j | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | metnrm 24866 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm) |
3 | 1 | methaus 24517 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
4 | haust1 23344 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Fre) |
6 | nrmreg 23816 | . 2 ⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg) | |
7 | 2, 5, 6 | syl2anc 582 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 ∞Metcxmet 21324 MetOpencmopn 21329 Frect1 23299 Hauscha 23300 Regcreg 23301 Nrmcnrm 23302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-ec 8728 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-n0 12519 df-z 12605 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-icc 13379 df-topgen 17453 df-qtop 17517 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-top 22884 df-topon 22901 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-cn 23219 df-t0 23305 df-t1 23306 df-haus 23307 df-reg 23308 df-nrm 23309 df-kq 23686 df-hmeo 23747 df-hmph 23748 |
This theorem is referenced by: (None) |
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