![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24406 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
3 | 1 | tmsbas 24405 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
4 | 3 | fveq2d 6901 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∞Met‘𝑋) = (∞Met‘(Base‘𝐾))) |
5 | 2, 4 | eleq12d 2823 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)))) |
6 | 5 | ibi 267 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝐾) ∈ (∞Met‘(Base‘𝐾))) |
7 | ssid 4002 | . . 3 ⊢ (Base‘𝐾) ⊆ (Base‘𝐾) | |
8 | xmetres2 24280 | . . 3 ⊢ (((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) ∧ (Base‘𝐾) ⊆ (Base‘𝐾)) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) | |
9 | 6, 7, 8 | sylancl 585 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾))) |
10 | xmetf 24248 | . . . . . 6 ⊢ ((dist‘𝐾) ∈ (∞Met‘(Base‘𝐾)) → (dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ*) | |
11 | ffn 6722 | . . . . . 6 ⊢ ((dist‘𝐾):((Base‘𝐾) × (Base‘𝐾))⟶ℝ* → (dist‘𝐾) Fn ((Base‘𝐾) × (Base‘𝐾))) | |
12 | fnresdm 6674 | . . . . . 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 2771 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
15 | 14 | fveq2d 6901 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘𝐷)) |
16 | eqid 2728 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
17 | 1, 16 | tmstopn 24407 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
18 | 15, 17 | eqtr2d 2769 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) |
19 | eqid 2728 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
20 | eqid 2728 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2728 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
22 | 19, 20, 21 | isxms2 24367 | . 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 582 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 × cxp 5676 ↾ cres 5680 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 ℝ*cxr 11278 Basecbs 17180 distcds 17242 TopOpenctopn 17403 ∞Metcxmet 21264 MetOpencmopn 21269 ∞MetSpcxms 24236 toMetSpctms 24238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-tset 17252 df-ds 17255 df-rest 17404 df-topn 17405 df-topgen 17425 df-psmet 21271 df-xmet 21272 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-xms 24239 df-tms 24241 |
This theorem is referenced by: tmsms 24409 tmsxps 24458 tmsxpsmopn 24459 tmsxpsval 24460 |
Copyright terms: Public domain | W3C validator |