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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 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
metreg | β’ (π· β (βMetβπ) β π½ β Reg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metnrm.j | . . 3 β’ π½ = (MetOpenβπ·) | |
2 | 1 | metnrm 24700 | . 2 β’ (π· β (βMetβπ) β π½ β Nrm) |
3 | 1 | methaus 24351 | . . 3 β’ (π· β (βMetβπ) β π½ β Haus) |
4 | haust1 23178 | . . 3 β’ (π½ β Haus β π½ β Fre) | |
5 | 3, 4 | syl 17 | . 2 β’ (π· β (βMetβπ) β π½ β Fre) |
6 | nrmreg 23650 | . 2 β’ ((π½ β Nrm β§ π½ β Fre) β π½ β Reg) | |
7 | 2, 5, 6 | syl2anc 583 | 1 β’ (π· β (βMetβπ) β π½ β Reg) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6533 βMetcxmet 21213 MetOpencmopn 21218 Frect1 23133 Hauscha 23134 Regcreg 23135 Nrmcnrm 23136 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-ec 8701 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-icc 13328 df-topgen 17388 df-qtop 17452 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-top 22718 df-topon 22735 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-cn 23053 df-t0 23139 df-t1 23140 df-haus 23141 df-reg 23142 df-nrm 23143 df-kq 23520 df-hmeo 23581 df-hmph 23582 |
This theorem is referenced by: (None) |
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