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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 24153 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm) |
3 | 1 | methaus 23804 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
4 | haust1 22631 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Fre) |
6 | nrmreg 23103 | . 2 ⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg) | |
7 | 2, 5, 6 | syl2anc 585 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6492 ∞Metcxmet 20710 MetOpencmopn 20715 Frect1 22586 Hauscha 22587 Regcreg 22588 Nrmcnrm 22589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-ec 8584 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-sup 9312 df-inf 9313 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-n0 12348 df-z 12434 df-uz 12698 df-q 12804 df-rp 12846 df-xneg 12963 df-xadd 12964 df-xmul 12965 df-icc 13201 df-topgen 17261 df-qtop 17325 df-psmet 20717 df-xmet 20718 df-met 20719 df-bl 20720 df-mopn 20721 df-top 22171 df-topon 22188 df-bases 22224 df-cld 22298 df-ntr 22299 df-cls 22300 df-cn 22506 df-t0 22592 df-t1 22593 df-haus 22594 df-reg 22595 df-nrm 22596 df-kq 22973 df-hmeo 23034 df-hmph 23035 |
This theorem is referenced by: (None) |
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