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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 28993 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | hlcmet.x | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hlcmet.8 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
4 | 2, 3 | cbncms 28970 | . 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 1543 ∈ wcel 2111 ‘cfv 6398 CMetccmet 24175 BaseSetcba 28691 IndMetcims 28696 CBanccbn 28967 CHilOLDchlo 28990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-iota 6356 df-fv 6406 df-cbn 28968 df-hlo 28991 |
This theorem is referenced by: hlmet 29000 hlcompl 29020 |
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