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Theorem isxms2 15446

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.

Step	Hyp	Ref	Expression
1		isms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
2		isms.x	. . 3 ⊢ 𝑋 = (Base‘𝐾)
3		isms.d	. . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
4	1, 2, 3	isxms 15445	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
5	2, 1	istps 15026	. . . 4 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
6		df-mopn 14824	. . . . . . . . . 10 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥)))
7	6	dmmptss 5264	. . . . . . . . 9 ⊢ dom MetOpen ⊆ ∪ ran ∞Met
8		mopnrel 15435	. . . . . . . . . 10 ⊢ Rel MetOpen
9		toponmax 15019	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
10	9	adantl 277	. . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ 𝐽)
11		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 = (MetOpen‘𝐷))
12	10, 11	eleqtrd 2313	. . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ (MetOpen‘𝐷))
13		relelfvdm 5707	. . . . . . . . . 10 ⊢ ((Rel MetOpen ∧ 𝑋 ∈ (MetOpen‘𝐷)) → 𝐷 ∈ dom MetOpen)
14	8, 12, 13	sylancr 414	. . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ dom MetOpen)
15	7, 14	sselid 3240	. . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ ∪ ran ∞Met)
16		xmetunirn 15352	. . . . . . . 8 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
17	15, 16	sylib 122	. . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
18		eqid 2234	. . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
19	18	mopntopon 15437	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷))
20	17, 19	syl 14	. . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷))
21	11, 20	eqeltrd 2311	. . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘dom dom 𝐷))
22		toponuni 15009	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘dom dom 𝐷) → dom dom 𝐷 = ∪ 𝐽)
23	21, 22	syl 14	. . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = ∪ 𝐽)
24		toponuni 15009	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
25	24	adantl 277	. . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 = ∪ 𝐽)
26	23, 25	eqtr4d 2270	. . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = 𝑋)
27	26	fveq2d 5679	. . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∞Met‘dom dom 𝐷) = (∞Met‘𝑋))
28	17, 27	eleqtrd 2313	. . . . . 6 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
29	28	ex 115	. . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)))
30	18	mopntopon 15437	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ (TopOn‘𝑋))
31		eleq1 2297	. . . . . 6 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ (MetOpen‘𝐷) ∈ (TopOn‘𝑋)))
32	30, 31	imbitrrid 156	. . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)))
33	29, 32	impbid 129	. . . 4 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐷 ∈ (∞Met‘𝑋)))
34	5, 33	bitrid 192	. . 3 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐾 ∈ TopSp ↔ 𝐷 ∈ (∞Met‘𝑋)))
35	34	pm5.32ri 455	. 2 ⊢ ((𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
36	4, 35	bitri 184	1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2205 ∪ cuni 3919 × cxp 4752 dom cdm 4754 ran crn 4755 ↾ cres 4756 Rel wrel 4759 ‘cfv 5357 Basecbs 13299 distcds 13386 TopOpenctopn 13540 topGenctg 13554 ∞Metcxmet 14813 ballcbl 14815 MetOpencmopn 14818 TopOnctopon 15004 TopSpctps 15024 ∞MetSpcxms 15330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-map 6897 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-5 9319 df-6 9320 df-7 9321 df-8 9322 df-9 9323 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-xneg 10127 df-xadd 10128 df-seqfrec 10837 df-exp 10928 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712 df-ndx 13302 df-slot 13303 df-base 13305 df-tset 13396 df-rest 13541 df-topn 13542 df-topgen 13560 df-psmet 14820 df-xmet 14821 df-bl 14823 df-mopn 14824 df-top 14992 df-topon 15005 df-topsp 15025 df-bases 15037 df-xms 15333

This theorem is referenced by: isms2 15448 xmsxmet 15454

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