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Theorem cmsss 25227

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 5646	. . . . . . 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 5968	. . . . 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 6854	. . . . . . . . 9 ⊢ 𝑋 ∈ V
7	6	ssex 5271	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (dist‘𝑀) = (dist‘𝑀)
11	9, 10	ressds 17349	. . . . . . 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 17184	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 = (Base‘𝐾))
14	13	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (Base‘𝐾))
15	14	sqxpeqd 5663	. . . . . 6 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) = ((Base‘𝐾) × (Base‘𝐾)))
16	12, 15	reseq12d 5940	. . . . 5 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((dist‘𝑀) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
17	4, 16	eqtrd 2764	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
18	14	fveq2d 6844	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (CMet‘𝐴) = (CMet‘(Base‘𝐾)))
19	17, 18	eleq12d 2822	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) ∈ (CMet‘𝐴) ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
20		eqid 2729	. . . . . 6 ⊢ ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
21	5, 20	cmscmet 25222	. . . . 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 2729	. . . . 5 ⊢ (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))) = (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋)))
24	23	cmetss 25192	. . . 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 25224	. . . 4 ⊢ (𝑀 ∈ CMetSp → 𝑀 ∈ MetSp)
28		ressms 24390	. . . . 5 ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ V) → (𝑀 ↾_s 𝐴) ∈ MetSp)
29	9, 28	eqeltrid 2832	. . . 4 ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ V) → 𝐾 ∈ MetSp)
30	27, 7, 29	syl2an 596	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ MetSp)
31		eqid 2729	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
32		eqid 2729	. . . . 5 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
33	31, 32	iscms 25221	. . . 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 24316	. . . . 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 6844	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (Clsd‘𝐽) = (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋)))))
41	40	eleq2d 2814	. 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 1540 ∈ wcel 2109 Vcvv 3444 ⊆ wss 3911 × cxp 5629 ↾ cres 5633 ‘cfv 6499 (class class class)co 7369 Basecbs 17155 ↾_s cress 17176 distcds 17205 TopOpenctopn 17360 MetOpencmopn 21230 Clsdccld 22879 MetSpcms 24182 CMetccmet 25130 CMetSpccms 25208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ico 13288 df-icc 13289 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-tset 17215 df-ds 17218 df-rest 17361 df-topn 17362 df-topgen 17382 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-haus 23178 df-fil 23709 df-flim 23802 df-xms 24184 df-ms 24185 df-cfil 25131 df-cmet 25133 df-cms 25211

This theorem is referenced by: lssbn 25228 resscdrg 25234 srabn 25236 ishl2 25246 recms 25256 pjthlem2 25314

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