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| Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25373 instead. (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| iscbn.x | ⊢ 𝑋 = (BaseSet‘𝑈) | 
| iscbn.8 | ⊢ 𝐷 = (IndMet‘𝑈) | 
| Ref | Expression | 
|---|---|
| iscbn | ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈)) | |
| 2 | iscbn.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
| 3 | 1, 2 | eqtr4di 2794 | . . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷) | 
| 4 | fveq2 6905 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
| 5 | iscbn.x | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 6 | 4, 5 | eqtr4di 2794 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) | 
| 7 | 6 | fveq2d 6909 | . . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋)) | 
| 8 | 3, 7 | eleq12d 2834 | . 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋))) | 
| 9 | df-cbn 30883 | . 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))} | |
| 10 | 8, 9 | elrab2 3694 | 1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = 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: cbncms 30885 bnnv 30886 bnsscmcl 30888 cnbn 30889 hhhl 31224 hhssbnOLD 31299 | 
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