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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 27720 | . 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) |
4 | 3 | simprbi 480 | 1 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 CMetcms 23052 NrmCVeccnv 27439 BaseSetcba 27441 IndMetcims 27446 CBanccbn 27718 |
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-cbn 27719 |
This theorem is referenced by: bnsscmcl 27724 ubthlem1 27726 ubthlem2 27727 minvecolem4a 27733 hlcmet 27750 |
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