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Mirrors > Home > MPE Home > Th. List > setsxms | 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 |
---|---|
setsxms | ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsms.x | . . . . 5 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
2 | setsms.d | . . . . 5 ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) | |
3 | setsms.k | . . . . 5 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
4 | setsms.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
5 | 1, 2, 3, 4 | setsmstopn 23082 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
6 | 1, 2, 3 | setsmsds 23080 | . . . . . . 7 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
7 | 1, 2, 3 | setsmsbas 23079 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
8 | 7 | sqxpeqd 5582 | . . . . . . 7 ⊢ (𝜑 → (𝑋 × 𝑋) = ((Base‘𝐾) × (Base‘𝐾))) |
9 | 6, 8 | reseq12d 5849 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
10 | 2, 9 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → 𝐷 = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
11 | 10 | fveq2d 6669 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) |
12 | 5, 11 | eqtr3d 2858 | . . 3 ⊢ (𝜑 → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) |
13 | eqid 2821 | . . . . 5 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
14 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
15 | eqid 2821 | . . . . 5 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
16 | 13, 14, 15 | isxms2 23052 | . . . 4 ⊢ (𝐾 ∈ ∞MetSp ↔ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)) ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))) |
17 | 16 | rbaib 541 | . . 3 ⊢ ((TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) → (𝐾 ∈ ∞MetSp ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))) |
18 | 12, 17 | syl 17 | . 2 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))) |
19 | 7 | fveq2d 6669 | . . 3 ⊢ (𝜑 → (∞Met‘𝑋) = (∞Met‘(Base‘𝐾))) |
20 | 10, 19 | eleq12d 2907 | . 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ↔ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))) |
21 | 18, 20 | bitr4d 284 | 1 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 〈cop 4567 × cxp 5548 ↾ cres 5552 ‘cfv 6350 (class class class)co 7150 ndxcnx 16474 sSet csts 16475 Basecbs 16477 TopSetcts 16565 distcds 16568 TopOpenctopn 16689 ∞Metcxmet 20524 MetOpencmopn 20529 ∞MetSpcxms 22921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-tset 16578 df-ds 16581 df-rest 16690 df-topn 16691 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 |
This theorem is referenced by: setsms 23084 |
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