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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 6660 | . . 3 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) ∈ V) | |
2 | fveq2 6645 | . . . . . . 7 ⊢ (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀)) | |
3 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀)) |
4 | id 22 | . . . . . . . 8 ⊢ (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤)) | |
5 | fveq2 6645 | . . . . . . . . 9 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀)) | |
6 | iscms.1 | . . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑀) | |
7 | 5, 6 | eqtr4di 2851 | . . . . . . . 8 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋) |
8 | 4, 7 | sylan9eqr 2855 | . . . . . . 7 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋) |
9 | 8 | sqxpeqd 5551 | . . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋)) |
10 | 3, 9 | reseq12d 5819 | . . . . 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 2851 | . . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷) |
13 | 8 | fveq2d 6649 | . . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋)) |
14 | 12, 13 | eleq12d 2884 | . . 3 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋))) |
15 | 1, 14 | sbcied 3762 | . 2 ⊢ (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋))) |
16 | df-cms 23939 | . 2 ⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)} | |
17 | 15, 16 | elrab2 3631 | 1 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 × cxp 5517 ↾ cres 5521 ‘cfv 6324 Basecbs 16475 distcds 16566 MetSpcms 22925 CMetccmet 23858 CMetSpccms 23936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-res 5531 df-iota 6283 df-fv 6332 df-cms 23939 |
This theorem is referenced by: cmscmet 23950 cmsms 23952 cmspropd 23953 cmssmscld 23954 cmsss 23955 cncms 23959 cmscsscms 23977 cssbn 23979 |
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