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Theorem cmsss 25386

Description: The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Hypotheses**
Ref	Expression
cmsss.h	⊢ 𝐾 = (𝑀 ↾_s 𝐴)
cmsss.x	⊢ 𝑋 = (Base‘𝑀)
cmsss.j	⊢ 𝐽 = (TopOpen‘𝑀)

**Assertion**
Ref	Expression
cmsss	⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽)))

**Proof of Theorem cmsss**
Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
2		xpss12 5699	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
3	1, 2	sylancom 588	. . . . . 6 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
4	3	resabs1d 6025	. . . . 5 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) = ((dist‘𝑀) ↾ (𝐴 × 𝐴)))
5		cmsss.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑀)
6	5	fvexi 6919	. . . . . . . . 9 ⊢ 𝑋 ∈ V
7	6	ssex 5320	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ∈ V)
8	7	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
9		cmsss.h	. . . . . . . 8 ⊢ 𝐾 = (𝑀 ↾_s 𝐴)
10		eqid 2736	. . . . . . . 8 ⊢ (dist‘𝑀) = (dist‘𝑀)
11	9, 10	ressds 17455	. . . . . . 7 ⊢ (𝐴 ∈ V → (dist‘𝑀) = (dist‘𝐾))
12	8, 11	syl 17	. . . . . 6 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (dist‘𝑀) = (dist‘𝐾))
13	9, 5	ressbas2 17284	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 = (Base‘𝐾))
14	13	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (Base‘𝐾))
15	14	sqxpeqd 5716	. . . . . 6 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) = ((Base‘𝐾) × (Base‘𝐾)))
16	12, 15	reseq12d 5997	. . . . 5 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((dist‘𝑀) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
17	4, 16	eqtrd 2776	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
18	14	fveq2d 6909	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (CMet‘𝐴) = (CMet‘(Base‘𝐾)))
19	17, 18	eleq12d 2834	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) ∈ (CMet‘𝐴) ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
20		eqid 2736	. . . . . 6 ⊢ ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
21	5, 20	cmscmet 25381	. . . . 5 ⊢ (𝑀 ∈ CMetSp → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ∈ (CMet‘𝑋))
22	21	adantr 480	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ∈ (CMet‘𝑋))
23		eqid 2736	. . . . 5 ⊢ (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))) = (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋)))
24	23	cmetss 25351	. . . 4 ⊢ (((dist‘𝑀) ↾ (𝑋 × 𝑋)) ∈ (CMet‘𝑋) → ((((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) ∈ (CMet‘𝐴) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
25	22, 24	syl 17	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) ∈ (CMet‘𝐴) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
26	19, 25	bitr3d 281	. 2 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
27		cmsms 25383	. . . 4 ⊢ (𝑀 ∈ CMetSp → 𝑀 ∈ MetSp)
28		ressms 24540	. . . . 5 ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ V) → (𝑀 ↾_s 𝐴) ∈ MetSp)
29	9, 28	eqeltrid 2844	. . . 4 ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ V) → 𝐾 ∈ MetSp)
30	27, 7, 29	syl2an 596	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ MetSp)
31		eqid 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
32		eqid 2736	. . . . 5 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
33	31, 32	iscms 25380	. . . 4 ⊢ (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
34	33	baib 535	. . 3 ⊢ (𝐾 ∈ MetSp → (𝐾 ∈ CMetSp ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
35	30, 34	syl 17	. 2 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐾 ∈ CMetSp ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
36	27	adantr 480	. . . . 5 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝑀 ∈ MetSp)
37		cmsss.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑀)
38	37, 5, 20	mstopn 24463	. . . . 5 ⊢ (𝑀 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))
39	36, 38	syl 17	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐽 = (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))
40	39	fveq2d 6909	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (Clsd‘𝐽) = (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋)))))
41	40	eleq2d 2826	. 2 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
42	26, 35, 41	3bitr4d 311	1 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 × cxp 5682 ↾ cres 5686 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾_s cress 17275 distcds 17307 TopOpenctopn 17467 MetOpencmopn 21355 Clsdccld 23025 MetSpcms 24329 CMetccmet 25289 CMetSpccms 25367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fi 9452 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13394 df-icc 13395 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-tset 17317 df-ds 17320 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-haus 23324 df-fil 23855 df-flim 23948 df-xms 24331 df-ms 24332 df-cfil 25290 df-cmet 25292 df-cms 25370

This theorem is referenced by: lssbn 25387 resscdrg 25393 srabn 25395 ishl2 25405 recms 25415 pjthlem2 25473

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