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Theorem cmsss 23956

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 487	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
2		xpss12 5572	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
3	1, 2	sylancom 590	. . . . . 6 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
4	3	resabs1d 5886	. . . . 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 6686	. . . . . . . . 9 ⊢ 𝑋 ∈ V
7	6	ssex 5227	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ∈ V)
8	7	adantl 484	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
9		cmsss.h	. . . . . . . 8 ⊢ 𝐾 = (𝑀 ↾_s 𝐴)
10		eqid 2823	. . . . . . . 8 ⊢ (dist‘𝑀) = (dist‘𝑀)
11	9, 10	ressds 16688	. . . . . . 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 16557	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 = (Base‘𝐾))
14	13	adantl 484	. . . . . . 7 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (Base‘𝐾))
15	14	sqxpeqd 5589	. . . . . 6 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) = ((Base‘𝐾) × (Base‘𝐾)))
16	12, 15	reseq12d 5856	. . . . 5 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((dist‘𝑀) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
17	4, 16	eqtrd 2858	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
18	14	fveq2d 6676	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (CMet‘𝐴) = (CMet‘(Base‘𝐾)))
19	17, 18	eleq12d 2909	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((((dist‘𝑀) ↾ (𝑋 × 𝑋)) ↾ (𝐴 × 𝐴)) ∈ (CMet‘𝐴) ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
20		eqid 2823	. . . . . 6 ⊢ ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
21	5, 20	cmscmet 23951	. . . . 5 ⊢ (𝑀 ∈ CMetSp → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ∈ (CMet‘𝑋))
22	21	adantr 483	. . . 4 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) ∈ (CMet‘𝑋))
23		eqid 2823	. . . . 5 ⊢ (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))) = (MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋)))
24	23	cmetss 23921	. . . 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 283	. 2 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
27		cmsms 23953	. . . 4 ⊢ (𝑀 ∈ CMetSp → 𝑀 ∈ MetSp)
28		ressms 23138	. . . . 5 ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ V) → (𝑀 ↾_s 𝐴) ∈ MetSp)
29	9, 28	eqeltrid 2919	. . . 4 ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ V) → 𝐾 ∈ MetSp)
30	27, 7, 29	syl2an 597	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ MetSp)
31		eqid 2823	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
32		eqid 2823	. . . . 5 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
33	31, 32	iscms 23950	. . . 4 ⊢ (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
34	33	baib 538	. . 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 483	. . . . 5 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → 𝑀 ∈ MetSp)
37		cmsss.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑀)
38	37, 5, 20	mstopn 23064	. . . . 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 6676	. . 3 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (Clsd‘𝐽) = (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋)))))
41	40	eleq2d 2900	. 2 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘(MetOpen‘((dist‘𝑀) ↾ (𝑋 × 𝑋))))))
42	26, 35, 41	3bitr4d 313	1 ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 × cxp 5555 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾_s cress 16486 distcds 16576 TopOpenctopn 16697 MetOpencmopn 20537 Clsdccld 21626 MetSpcms 22930 CMetccmet 23859 CMetSpccms 23937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-icc 12748 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-tset 16586 df-ds 16589 df-rest 16698 df-topn 16699 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-haus 21925 df-fil 22456 df-flim 22549 df-xms 22932 df-ms 22933 df-cfil 23860 df-cmet 23862 df-cms 23940

This theorem is referenced by: lssbn 23957 resscdrg 23963 srabn 23965 ishl2 23975 recms 23985 pjthlem2 24043

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