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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 25093 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | cphlvec 25096 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | |
3 | isnvc 24605 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
4 | 1, 2, 3 | sylanbrc 582 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 LVecclvec 20980 NrmModcnlm 24482 NrmVeccnvc 24483 ℂPreHilccph 25087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fv 6550 df-ov 7417 df-phl 21551 df-nvc 24489 df-cph 25089 |
This theorem is referenced by: ishl2 25291 csschl 25297 |
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