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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.) Use isbn 23941 instead. (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 6670 | . . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈)) | |
2 | iscbn.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷) |
4 | fveq2 6670 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
5 | iscbn.x | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | 4, 5 | syl6eqr 2874 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
7 | 6 | fveq2d 6674 | . . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋)) |
8 | 3, 7 | eleq12d 2907 | . 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋))) |
9 | df-cbn 28640 | . 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))} | |
10 | 8, 9 | elrab2 3683 | 1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 CMetccmet 23857 NrmCVeccnv 28361 BaseSetcba 28363 IndMetcims 28368 CBanccbn 28639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-cbn 28640 |
This theorem is referenced by: cbncms 28642 bnnv 28643 bnsscmcl 28645 cnbn 28646 hhhl 28981 hhssbnOLD 29056 |
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