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Mirrors > Home > MPE Home > Th. List > setsms | Structured version Visualization version GIF version |
Description: The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
setsms.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
Ref | Expression |
---|---|
setsms | ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsms.x | . . . 4 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
2 | setsms.d | . . . 4 ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) | |
3 | setsms.k | . . . 4 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
4 | setsms.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
5 | 1, 2, 3, 4 | setsxms 23832 | . . 3 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋))) |
6 | 1, 2, 3 | setsmsds 23828 | . . . . . 6 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
7 | 1, 2, 3 | setsmsbas 23826 | . . . . . . 7 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
8 | 7 | sqxpeqd 5665 | . . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑋) = ((Base‘𝐾) × (Base‘𝐾))) |
9 | 6, 8 | reseq12d 5938 | . . . . 5 ⊢ (𝜑 → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
10 | 2, 9 | eqtr2d 2777 | . . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
11 | 7 | fveq2d 6846 | . . . . 5 ⊢ (𝜑 → (Met‘𝑋) = (Met‘(Base‘𝐾))) |
12 | 11 | eqcomd 2742 | . . . 4 ⊢ (𝜑 → (Met‘(Base‘𝐾)) = (Met‘𝑋)) |
13 | 10, 12 | eleq12d 2832 | . . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ 𝐷 ∈ (Met‘𝑋))) |
14 | 5, 13 | anbi12d 631 | . 2 ⊢ (𝜑 → ((𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (Met‘𝑋)))) |
15 | eqid 2736 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
16 | eqid 2736 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2736 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
18 | 15, 16, 17 | isms 23800 | . 2 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)))) |
19 | metxmet 23685 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
20 | 19 | pm4.71ri 561 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (Met‘𝑋))) |
21 | 14, 18, 20 | 3bitr4g 313 | 1 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4592 × cxp 5631 ↾ cres 5635 ‘cfv 6496 (class class class)co 7356 sSet csts 17034 ndxcnx 17064 Basecbs 17082 TopSetcts 17138 distcds 17141 TopOpenctopn 17302 ∞Metcxmet 20779 Metcmet 20780 MetOpencmopn 20784 ∞MetSpcxms 23668 MetSpcms 23669 |
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 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
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-iun 4956 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 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-sup 9377 df-inf 9378 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-tset 17151 df-ds 17154 df-rest 17303 df-topn 17304 df-topgen 17324 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-xms 23671 df-ms 23672 |
This theorem is referenced by: (None) |
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