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Mirrors > Home > MPE Home > Th. List > iscbn | Structured version Visualization version GIF version |
Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 24754 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 6862 | . . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈)) | |
2 | iscbn.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
3 | 1, 2 | eqtr4di 2789 | . . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷) |
4 | fveq2 6862 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
5 | iscbn.x | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | 4, 5 | eqtr4di 2789 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
7 | 6 | fveq2d 6866 | . . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋)) |
8 | 3, 7 | eleq12d 2826 | . 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋))) |
9 | df-cbn 29902 | . 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))} | |
10 | 8, 9 | elrab2 3666 | 1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6516 CMetccmet 24670 NrmCVeccnv 29623 BaseSetcba 29625 IndMetcims 29630 CBanccbn 29901 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-iota 6468 df-fv 6524 df-cbn 29902 |
This theorem is referenced by: cbncms 29904 bnnv 29905 bnsscmcl 29907 cnbn 29908 hhhl 30243 hhssbnOLD 30318 |
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