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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 3796 | . . 3 ⊢ (𝑊 ∈ (NrmVec ∩ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp)) | |
2 | 1 | anbi1i 731 | . 2 ⊢ ((𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp)) |
3 | fveq2 6191 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) | |
4 | isbn.1 | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | 3, 4 | syl6eqr 2674 | . . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹) |
6 | 5 | eleq1d 2686 | . . 3 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ CMetSp ↔ 𝐹 ∈ CMetSp)) |
7 | df-bn 23133 | . . 3 ⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp} | |
8 | 6, 7 | elrab2 3366 | . 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp)) |
9 | df-3an 1039 | . 2 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp)) | |
10 | 2, 8, 9 | 3bitr4i 292 | 1 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ‘cfv 5888 Scalarcsca 15944 NrmVeccnvc 22386 CMetSpccms 23129 Bancbn 23130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-bn 23133 |
This theorem is referenced by: bnsca 23136 bnnvc 23137 bncms 23141 lssbn 23148 srabn 23156 ishl2 23166 |
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