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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 23021 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
3 | 1 | tmsbas 23020 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
4 | 3 | fveq2d 6667 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∞Met‘𝑋) = (∞Met‘(Base‘𝐾))) |
5 | 2, 4 | eleq12d 2904 | . . . 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 3986 | . . 3 ⊢ (Base‘𝐾) ⊆ (Base‘𝐾) | |
8 | xmetres2 22898 | . . 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 22866 | . . . . . 6 ⊢ ((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) → (dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ*) | |
11 | ffn 6507 | . . . . . 6 ⊢ ((dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ* → (dist‘𝐾) Fn ((Base‘𝐾) × (Base‘𝐾))) | |
12 | fnresdm 6459 | . . . . . 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 2856 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
15 | 14 | fveq2d 6667 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘𝐷)) |
16 | eqid 2818 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
17 | 1, 16 | tmstopn 23022 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
18 | 15, 17 | eqtr2d 2854 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) |
19 | eqid 2818 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
20 | eqid 2818 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2818 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
22 | 19, 20, 21 | isxms2 22985 | . 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 3933 × cxp 5546 ↾ cres 5550 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 ℝ*cxr 10662 Basecbs 16471 distcds 16562 TopOpenctopn 16683 ∞Metcxmet 20458 MetOpencmopn 20463 ∞MetSpcxms 22854 toMetSpctms 22856 |
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 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-tset 16572 df-ds 16575 df-rest 16684 df-topn 16685 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-xms 22857 df-tms 22859 |
This theorem is referenced by: tmsms 23024 tmsxps 23073 tmsxpsmopn 23074 tmsxpsval 23075 |
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