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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 6921 | . . 3 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) ∈ V) | |
| 2 | fveq2 6906 | . . . . . . 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 6906 | . . . . . . . . 9 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀)) | |
| 6 | iscms.1 | . . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑀) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . . . . 8 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋) |
| 8 | 4, 7 | sylan9eqr 2799 | . . . . . . 7 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋) |
| 9 | 8 | sqxpeqd 5717 | . . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋)) |
| 10 | 3, 9 | reseq12d 5998 | . . . . 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 2795 | . . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷) |
| 13 | 8 | fveq2d 6910 | . . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋)) |
| 14 | 12, 13 | eleq12d 2835 | . . 3 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋))) |
| 15 | 1, 14 | sbcied 3832 | . 2 ⊢ (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋))) |
| 16 | df-cms 25369 | . 2 ⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)} | |
| 17 | 15, 16 | elrab2 3695 | 1 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 × cxp 5683 ↾ cres 5687 ‘cfv 6561 Basecbs 17247 distcds 17306 MetSpcms 24328 CMetccmet 25288 CMetSpccms 25366 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-cms 25369 |
| This theorem is referenced by: cmscmet 25380 cmsms 25382 cmspropd 25383 cmssmscld 25384 cmsss 25385 cncms 25389 cmscsscms 25407 cssbn 25409 |
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