Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hlcmet | Structured version Visualization version GIF version |
Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlcmet.x | ⊢ 𝑋 = (BaseSet‘𝑈) |
hlcmet.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
hlcmet | ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlobn 28667 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | hlcmet.x | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hlcmet.8 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
4 | 2, 3 | cbncms 28644 | . 2 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 CMetccmet 23859 BaseSetcba 28365 IndMetcims 28370 CBanccbn 28641 CHilOLDchlo 28664 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-cbn 28642 df-hlo 28665 |
This theorem is referenced by: hlmet 28674 hlcompl 28694 |
Copyright terms: Public domain | W3C validator |