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| Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25382 (or preferably bncms 25379) 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 30884 | . 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 1539 ∈ wcel 2107 ‘cfv 6560 CMetccmet 25289 NrmCVeccnv 30604 BaseSetcba 30606 IndMetcims 30611 CBanccbn 30882 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-cbn 30883 | 
| This theorem is referenced by: bnsscmcl 30888 ubthlem1 30890 ubthlem2 30891 minvecolem4a 30897 hlcmet 30914 | 
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