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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 23097 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
3 | 1 | tmsbas 23096 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
4 | 3 | fveq2d 6677 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∞Met‘𝑋) = (∞Met‘(Base‘𝐾))) |
5 | 2, 4 | eleq12d 2910 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)))) |
6 | 5 | ibi 269 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾))) |
7 | ssid 3992 | . . 3 ⊢ (Base‘𝐾) ⊆ (Base‘𝐾) | |
8 | xmetres2 22974 | . . 3 ⊢ (((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) ∧ (Base‘𝐾) ⊆ (Base‘𝐾)) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) | |
9 | 6, 7, 8 | sylancl 588 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) |
10 | xmetf 22942 | . . . . . 6 ⊢ ((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) → (dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ*) | |
11 | ffn 6517 | . . . . . 6 ⊢ ((dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ* → (dist‘𝐾) Fn ((Base‘𝐾) × (Base‘𝐾))) | |
12 | fnresdm 6469 | . . . . . 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 2862 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
15 | 14 | fveq2d 6677 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘𝐷)) |
16 | eqid 2824 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
17 | 1, 16 | tmstopn 23098 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
18 | 15, 17 | eqtr2d 2860 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) |
19 | eqid 2824 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
20 | eqid 2824 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2824 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
22 | 19, 20, 21 | isxms2 23061 | . 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 585 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 × cxp 5556 ↾ cres 5560 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 ℝ*cxr 10677 Basecbs 16486 distcds 16577 TopOpenctopn 16698 ∞Metcxmet 20533 MetOpencmopn 20538 ∞MetSpcxms 22930 toMetSpctms 22932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 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-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-tset 16587 df-ds 16590 df-rest 16699 df-topn 16700 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-xms 22933 df-tms 22935 |
This theorem is referenced by: tmsms 23100 tmsxps 23149 tmsxpsmopn 23150 tmsxpsval 23151 |
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