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Mirrors > Home > MPE Home > Th. List > hlnv | Structured version Visualization version GIF version |
Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlnv | ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlobn 27744 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | bnnv 27722 | . 2 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 NrmCVeccnv 27439 CBanccbn 27718 CHilOLDchlo 27741 |
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 df-hlo 27742 |
This theorem is referenced by: hlnvi 27748 hlvc 27749 hladdf 27755 hlcom 27756 hlass 27757 hl0cl 27758 hladdid 27759 hlmulf 27760 hlmulid 27761 hlmulass 27762 hldi 27763 hldir 27764 hlmul0 27765 hlipf 27766 hlipcj 27767 hlipgt0 27770 |
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