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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 23858 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | cphlvec 23861 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | |
3 | isnvc 23382 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
4 | 1, 2, 3 | sylanbrc 587 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 LVecclvec 19927 NrmModcnlm 23267 NrmVeccnvc 23268 ℂPreHilccph 23852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-nul 5169 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-xp 5523 df-cnv 5525 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fv 6336 df-ov 7146 df-phl 20376 df-nvc 23274 df-cph 23854 |
This theorem is referenced by: ishl2 24055 csschl 24061 |
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