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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 28665 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | hlcmet.x | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hlcmet.8 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
4 | 2, 3 | cbncms 28642 | . 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 6355 CMetccmet 23857 BaseSetcba 28363 IndMetcims 28368 CBanccbn 28639 CHilOLDchlo 28662 |
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 df-hlo 28663 |
This theorem is referenced by: hlmet 28672 hlcompl 28692 |
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