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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.) (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 6191 | . . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈)) | |
2 | iscbn.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
3 | 1, 2 | syl6eqr 2674 | . . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷) |
4 | fveq2 6191 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
5 | iscbn.x | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | 4, 5 | syl6eqr 2674 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
7 | 6 | fveq2d 6195 | . . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋)) |
8 | 3, 7 | eleq12d 2695 | . 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋))) |
9 | df-cbn 27719 | . 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))} | |
10 | 8, 9 | elrab2 3366 | 1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = 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: cbncms 27721 bnnv 27722 bnsscmcl 27724 cnbn 27725 hhhl 28061 hhssbn 28137 |
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