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| Mirrors > Home > MPE Home > Th. List > isbn | Structured version Visualization version GIF version | ||
| Description: A Banach space is a normed vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.) |
| Ref | Expression |
|---|---|
| isbn.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
| Ref | Expression |
|---|---|
| isbn | ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elin 3967 | . . 3 ⊢ (𝑊 ∈ (NrmVec ∩ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp)) | |
| 2 | 1 | anbi1i 624 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp)) |
| 3 | fveq2 6906 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) | |
| 4 | isbn.1 | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹) |
| 6 | 5 | eleq1d 2826 | . . 3 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ CMetSp ↔ 𝐹 ∈ CMetSp)) |
| 7 | df-bn 25370 | . . 3 ⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp} | |
| 8 | 6, 7 | elrab2 3695 | . 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp)) |
| 9 | df-3an 1089 | . 2 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp)) | |
| 10 | 2, 8, 9 | 3bitr4i 303 | 1 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ‘cfv 6561 Scalarcsca 17300 NrmVeccnvc 24594 CMetSpccms 25366 Bancbn 25367 |
| 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 |
| 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-rab 3437 df-v 3482 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-iota 6514 df-fv 6569 df-bn 25370 |
| This theorem is referenced by: bnsca 25373 bnnvc 25374 bncms 25378 lssbn 25386 srabn 25394 ishl2 25404 cmslssbn 25406 |
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