| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 30907 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
| 2 | bnnv 30885 | . 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 2108 NrmCVeccnv 30603 CBanccbn 30881 CHilOLDchlo 30904 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-cbn 30882 df-hlo 30905 |
| This theorem is referenced by: hlnvi 30911 hlvc 30912 hladdf 30918 hlcom 30919 hlass 30920 hl0cl 30921 hladdid 30922 hlmulf 30923 hlmulid 30924 hlmulass 30925 hldi 30926 hldir 30927 hlmul0 30928 hlipf 30929 hlipcj 30930 hlipgt0 30933 |
| Copyright terms: Public domain | W3C validator |