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Theorem isxms2 23053
Description: Express the predicate "𝑋, 𝐷 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
isxms2 (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))

Proof of Theorem isxms2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isms.j . . 3 𝐽 = (TopOpen‘𝐾)
2 isms.x . . 3 𝑋 = (Base‘𝐾)
3 isms.d . . 3 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
41, 2, 3isxms 23052 . 2 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
52, 1istps 21537 . . . 4 (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
6 df-mopn 20085 . . . . . . . . . 10 MetOpen = (𝑥 ran ∞Met ↦ (topGen‘ran (ball‘𝑥)))
76dmmptss 6073 . . . . . . . . 9 dom MetOpen ⊆ ran ∞Met
8 toponmax 21529 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
98adantl 485 . . . . . . . . . . 11 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋𝐽)
10 simpl 486 . . . . . . . . . . 11 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 = (MetOpen‘𝐷))
119, 10eleqtrd 2916 . . . . . . . . . 10 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ (MetOpen‘𝐷))
12 elfvdm 6684 . . . . . . . . . 10 (𝑋 ∈ (MetOpen‘𝐷) → 𝐷 ∈ dom MetOpen)
1311, 12syl 17 . . . . . . . . 9 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ dom MetOpen)
147, 13sseldi 3940 . . . . . . . 8 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ran ∞Met)
15 xmetunirn 22942 . . . . . . . 8 (𝐷 ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
1614, 15sylib 221 . . . . . . 7 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
17 eqid 2822 . . . . . . . . . . . . 13 (MetOpen‘𝐷) = (MetOpen‘𝐷)
1817mopntopon 23044 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘dom dom 𝐷) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷))
1916, 18syl 17 . . . . . . . . . . 11 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷))
2010, 19eqeltrd 2914 . . . . . . . . . 10 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘dom dom 𝐷))
21 toponuni 21517 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘dom dom 𝐷) → dom dom 𝐷 = 𝐽)
2220, 21syl 17 . . . . . . . . 9 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = 𝐽)
23 toponuni 21517 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2423adantl 485 . . . . . . . . 9 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 = 𝐽)
2522, 24eqtr4d 2860 . . . . . . . 8 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = 𝑋)
2625fveq2d 6656 . . . . . . 7 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∞Met‘dom dom 𝐷) = (∞Met‘𝑋))
2716, 26eleqtrd 2916 . . . . . 6 ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
2827ex 416 . . . . 5 (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)))
2917mopntopon 23044 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ (TopOn‘𝑋))
30 eleq1 2901 . . . . . 6 (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ (MetOpen‘𝐷) ∈ (TopOn‘𝑋)))
3129, 30syl5ibr 249 . . . . 5 (𝐽 = (MetOpen‘𝐷) → (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)))
3228, 31impbid 215 . . . 4 (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐷 ∈ (∞Met‘𝑋)))
335, 32syl5bb 286 . . 3 (𝐽 = (MetOpen‘𝐷) → (𝐾 ∈ TopSp ↔ 𝐷 ∈ (∞Met‘𝑋)))
3433pm5.32ri 579 . 2 ((𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
354, 34bitri 278 1 (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2114   cuni 4813   × cxp 5530  dom cdm 5532  ran crn 5533  cres 5534  cfv 6334  Basecbs 16474  distcds 16565  TopOpenctopn 16686  topGenctg 16702  ∞Metcxmet 20074  ballcbl 20076  MetOpencmopn 20079  TopOnctopon 21513  TopSpctps 21535  ∞MetSpcxms 22922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-inf 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-topgen 16708  df-psmet 20081  df-xmet 20082  df-bl 20084  df-mopn 20085  df-top 21497  df-topon 21514  df-topsp 21536  df-bases 21549  df-xms 22925
This theorem is referenced by:  isms2  23055  xmsxmet  23061  setsxms  23084  tmsxms  23091  imasf1oxms  23094  ressxms  23130  prdsxms  23135
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