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Mirrors > Home > MPE Home > Th. List > iscms | Structured version Visualization version GIF version |
Description: A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
iscms.1 | ⊢ 𝑋 = (Base‘𝑀) |
iscms.2 | ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
iscms | ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6922 | . . 3 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) ∈ V) | |
2 | fveq2 6907 | . . . . . . 7 ⊢ (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀)) | |
3 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀)) |
4 | id 22 | . . . . . . . 8 ⊢ (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤)) | |
5 | fveq2 6907 | . . . . . . . . 9 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀)) | |
6 | iscms.1 | . . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑀) | |
7 | 5, 6 | eqtr4di 2793 | . . . . . . . 8 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋) |
8 | 4, 7 | sylan9eqr 2797 | . . . . . . 7 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋) |
9 | 8 | sqxpeqd 5721 | . . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋)) |
10 | 3, 9 | reseq12d 6001 | . . . . 5 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
11 | iscms.2 | . . . . 5 ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)) | |
12 | 10, 11 | eqtr4di 2793 | . . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷) |
13 | 8 | fveq2d 6911 | . . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋)) |
14 | 12, 13 | eleq12d 2833 | . . 3 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋))) |
15 | 1, 14 | sbcied 3837 | . 2 ⊢ (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋))) |
16 | df-cms 25383 | . 2 ⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)} | |
17 | 15, 16 | elrab2 3698 | 1 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 [wsbc 3791 × cxp 5687 ↾ cres 5691 ‘cfv 6563 Basecbs 17245 distcds 17307 MetSpcms 24344 CMetccmet 25302 CMetSpccms 25380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-res 5701 df-iota 6516 df-fv 6571 df-cms 25383 |
This theorem is referenced by: cmscmet 25394 cmsms 25396 cmspropd 25397 cmssmscld 25398 cmsss 25399 cncms 25403 cmscsscms 25421 cssbn 25423 |
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