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Mirrors > Home > MPE Home > Th. List > setsms | Structured version Visualization version GIF version |
Description: The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
setsms.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
Ref | Expression |
---|---|
setsms | ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsms.x | . . . 4 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
2 | setsms.d | . . . 4 ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) | |
3 | setsms.k | . . . 4 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
4 | setsms.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
5 | 1, 2, 3, 4 | setsxms 23091 | . . 3 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋))) |
6 | 1, 2, 3 | setsmsds 23088 | . . . . . 6 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
7 | 1, 2, 3 | setsmsbas 23087 | . . . . . . 7 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
8 | 7 | sqxpeqd 5589 | . . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑋) = ((Base‘𝐾) × (Base‘𝐾))) |
9 | 6, 8 | reseq12d 5856 | . . . . 5 ⊢ (𝜑 → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
10 | 2, 9 | eqtr2d 2859 | . . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
11 | 7 | fveq2d 6676 | . . . . 5 ⊢ (𝜑 → (Met‘𝑋) = (Met‘(Base‘𝐾))) |
12 | 11 | eqcomd 2829 | . . . 4 ⊢ (𝜑 → (Met‘(Base‘𝐾)) = (Met‘𝑋)) |
13 | 10, 12 | eleq12d 2909 | . . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ 𝐷 ∈ (Met‘𝑋))) |
14 | 5, 13 | anbi12d 632 | . 2 ⊢ (𝜑 → ((𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (Met‘𝑋)))) |
15 | eqid 2823 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
16 | eqid 2823 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2823 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
18 | 15, 16, 17 | isms 23061 | . 2 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)))) |
19 | metxmet 22946 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
20 | 19 | pm4.71ri 563 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (Met‘𝑋))) |
21 | 14, 18, 20 | 3bitr4g 316 | 1 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 ndxcnx 16482 sSet csts 16483 Basecbs 16485 TopSetcts 16573 distcds 16576 TopOpenctopn 16697 ∞Metcxmet 20532 Metcmet 20533 MetOpencmopn 20537 ∞MetSpcxms 22929 MetSpcms 22930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 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-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-tset 16586 df-ds 16589 df-rest 16698 df-topn 16699 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-ms 22933 |
This theorem is referenced by: (None) |
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