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Mirrors > Home > MPE Home > Th. List > tmsxms | Structured version Visualization version GIF version |
Description: The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsbas.k | ⊢ 𝐾 = (toMetSp‘𝐷) |
Ref | Expression |
---|---|
tmsxms | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmsbas.k | . . . . . 6 ⊢ 𝐾 = (toMetSp‘𝐷) | |
2 | 1 | tmsds 23023 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
3 | 1 | tmsbas 23022 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
4 | 3 | fveq2d 6668 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∞Met‘𝑋) = (∞Met‘(Base‘𝐾))) |
5 | 2, 4 | eleq12d 2907 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)))) |
6 | 5 | ibi 268 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾))) |
7 | ssid 3988 | . . 3 ⊢ (Base‘𝐾) ⊆ (Base‘𝐾) | |
8 | xmetres2 22900 | . . 3 ⊢ (((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) ∧ (Base‘𝐾) ⊆ (Base‘𝐾)) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) | |
9 | 6, 7, 8 | sylancl 586 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) |
10 | xmetf 22868 | . . . . . 6 ⊢ ((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) → (dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ*) | |
11 | ffn 6508 | . . . . . 6 ⊢ ((dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ* → (dist‘𝐾) Fn ((Base‘𝐾) × (Base‘𝐾))) | |
12 | fnresdm 6460 | . . . . . 6 ⊢ ((dist‘𝐾) Fn ((Base‘𝐾) × (Base‘𝐾)) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = (dist‘𝐾)) | |
13 | 6, 10, 11, 12 | 4syl 19 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = (dist‘𝐾)) |
14 | 13, 2 | eqtr4d 2859 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
15 | 14 | fveq2d 6668 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘𝐷)) |
16 | eqid 2821 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
17 | 1, 16 | tmstopn 23024 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
18 | 15, 17 | eqtr2d 2857 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) |
19 | eqid 2821 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
20 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2821 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
22 | 19, 20, 21 | isxms2 22987 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)) ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))) |
23 | 9, 18, 22 | sylanbrc 583 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 × cxp 5547 ↾ cres 5551 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 ℝ*cxr 10663 Basecbs 16473 distcds 16564 TopOpenctopn 16685 ∞Metcxmet 20460 MetOpencmopn 20465 ∞MetSpcxms 22856 toMetSpctms 22858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-fz 12883 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-tset 16574 df-ds 16577 df-rest 16686 df-topn 16687 df-topgen 16707 df-psmet 20467 df-xmet 20468 df-bl 20470 df-mopn 20471 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-xms 22859 df-tms 22861 |
This theorem is referenced by: tmsms 23026 tmsxps 23075 tmsxpsmopn 23076 tmsxpsval 23077 |
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