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Mirrors > Home > MPE Home > Th. List > cphnvc | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
cphnvc | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphnlm 23778 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | cphlvec 23781 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | |
3 | isnvc 23306 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
4 | 1, 2, 3 | sylanbrc 585 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 LVecclvec 19876 NrmModcnlm 23192 NrmVeccnvc 23193 ℂPreHilccph 23772 |
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 ax-nul 5212 |
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-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 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-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fv 6365 df-ov 7161 df-phl 20772 df-nvc 23199 df-cph 23774 |
This theorem is referenced by: ishl2 23975 csschl 23981 |
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