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Theorem cmsss 24715

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 485	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
2		xpss12 5648	. . . . . . 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 6856	. . . . . . . . 9 ⊢ 𝑋 ∈ V
7	6	ssex 5278	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ∈ V)
8	7	adantl 482	. . . . . . 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 17291	. . . . . . 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 17120	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 = (Base‘𝐾))
14	13	adantl 482	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (Base‘𝐾))
15	14	sqxpeqd 5665	. . . . . 6 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) = ((Base‘𝐾) × (Base‘𝐾)))
16	12, 15	reseq12d 5938	. . . . 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 6846	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (CMet‘𝐴) = (CMet‘(Base‘𝐾)))
19	17, 18	eleq12d 2832	. . 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 24710	. . . . 5 ⊢ (𝑀 ∈ CMetSp → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ∈ (CMet‘𝑋))
22	21	adantr 481	. . . 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 24680	. . . 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 280	. 2 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
27		cmsms 24712	. . . 4 ⊢ (𝑀 ∈ CMetSp → 𝑀 ∈ MetSp)
28		ressms 23882	. . . . 5 ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ V) → (𝑀 ↾_s 𝐴) ∈ MetSp)
29	9, 28	eqeltrid 2842	. . . 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 24709	. . . 4 ⊢ (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
34	33	baib 536	. . 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 481	. . . . 5 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝑀 ∈ MetSp)
37		cmsss.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑀)
38	37, 5, 20	mstopn 23805	. . . . 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 6846	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (Clsd‘𝐽) = (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋)))))
41	40	eleq2d 2823	. 2 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
42	26, 35, 41	3bitr4d 310	1 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 × cxp 5631 ↾ cres 5635 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 ↾_s cress 17112 distcds 17142 TopOpenctopn 17303 MetOpencmopn 20786 Clsdccld 22367 MetSpcms 23671 CMetccmet 24618 CMetSpccms 24696

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9347 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-icc 13271 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-tset 17152 df-ds 17155 df-rest 17304 df-topn 17305 df-topgen 17325 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-haus 22666 df-fil 23197 df-flim 23290 df-xms 23673 df-ms 23674 df-cfil 24619 df-cmet 24621 df-cms 24699

This theorem is referenced by: lssbn 24716 resscdrg 24722 srabn 24724 ishl2 24734 recms 24744 pjthlem2 24802

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