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Mirrors > Home > MPE Home > Th. List > cbncms | Structured version Visualization version GIF version |
Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (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 27104 | . 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) |
4 | 3 | simprbi 479 | 1 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 CMetcms 22860 NrmCVeccnv 26823 BaseSetcba 26825 IndMetcims 26830 CBanccbn 27102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-cbn 27103 |
This theorem is referenced by: bnsscmcl 27108 ubthlem1 27110 ubthlem2 27111 minvecolem4a 27117 hlcmet 27134 |
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