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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 22659 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
3 | 1 | tmsbas 22658 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
4 | 3 | fveq2d 6437 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∞Met‘𝑋) = (∞Met‘(Base‘𝐾))) |
5 | 2, 4 | eleq12d 2900 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)))) |
6 | 5 | ibi 259 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾))) |
7 | ssid 3848 | . . 3 ⊢ (Base‘𝐾) ⊆ (Base‘𝐾) | |
8 | xmetres2 22536 | . . 3 ⊢ (((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) ∧ (Base‘𝐾) ⊆ (Base‘𝐾)) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) | |
9 | 6, 7, 8 | sylancl 582 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) |
10 | xmetf 22504 | . . . . . 6 ⊢ ((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) → (dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ*) | |
11 | ffn 6278 | . . . . . 6 ⊢ ((dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ* → (dist‘𝐾) Fn ((Base‘𝐾) × (Base‘𝐾))) | |
12 | fnresdm 6233 | . . . . . 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 2864 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
15 | 14 | fveq2d 6437 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘𝐷)) |
16 | eqid 2825 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
17 | 1, 16 | tmstopn 22660 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
18 | 15, 17 | eqtr2d 2862 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) |
19 | eqid 2825 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
20 | eqid 2825 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2825 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
22 | 19, 20, 21 | isxms2 22623 | . 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 580 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 × cxp 5340 ↾ cres 5344 Fn wfn 6118 ⟶wf 6119 ‘cfv 6123 ℝ*cxr 10390 Basecbs 16222 distcds 16314 TopOpenctopn 16435 ∞Metcxmet 20091 MetOpencmopn 20096 ∞MetSpcxms 22492 toMetSpctms 22494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-tset 16324 df-ds 16327 df-rest 16436 df-topn 16437 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-bl 20101 df-mopn 20102 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-xms 22495 df-tms 22497 |
This theorem is referenced by: tmsms 22662 tmsxps 22711 tmsxpsmopn 22712 tmsxpsval 22713 |
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