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Mirrors > Home > MPE Home > Th. List > cbncms | Structured version Visualization version GIF version |
Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25108 (or preferably bncms 25105) instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
iscbn.x | ⊢ 𝑋 = (BaseSet‘𝑈) |
iscbn.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
cbncms | ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscbn.x | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | iscbn.8 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
3 | 1, 2 | iscbn 30399 | . 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) |
4 | 3 | simprbi 496 | 1 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 CMetccmet 25015 NrmCVeccnv 30119 BaseSetcba 30121 IndMetcims 30126 CBanccbn 30397 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-cbn 30398 |
This theorem is referenced by: bnsscmcl 30403 ubthlem1 30405 ubthlem2 30406 minvecolem4a 30412 hlcmet 30429 |
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