![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tmstopn | Structured version Visualization version GIF version |
Description: The topology of a constructed metric. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsbas.k | ⊢ 𝐾 = (toMetSp‘𝐷) |
tmstopn.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
tmstopn | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (TopOpen‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmstopn.j | . 2 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | eqid 2778 | . . . 4 ⊢ {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} | |
3 | tmsbas.k | . . . 4 ⊢ 𝐾 = (toMetSp‘𝐷) | |
4 | 2, 3 | tmslem 22699 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾))) |
5 | 4 | simp3d 1135 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
6 | 1, 5 | syl5eq 2826 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (TopOpen‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 {cpr 4400 〈cop 4404 ‘cfv 6137 ndxcnx 16256 Basecbs 16259 distcds 16351 TopOpenctopn 16472 ∞Metcxmet 20131 MetOpencmopn 20136 toMetSpctms 22536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-fz 12648 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-tset 16361 df-ds 16364 df-rest 16473 df-topn 16474 df-topgen 16494 df-psmet 20138 df-xmet 20139 df-bl 20141 df-mopn 20142 df-top 21110 df-topon 21127 df-bases 21162 df-tms 22539 |
This theorem is referenced by: tmsxms 22703 tmsxpsmopn 22754 |
Copyright terms: Public domain | W3C validator |