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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 23459 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | cphlvec 23462 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | |
3 | isnvc 22987 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
4 | 1, 2, 3 | sylanbrc 583 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 LVecclvec 19564 NrmModcnlm 22873 NrmVeccnvc 22874 ℂPreHilccph 23453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-nul 5101 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-xp 5449 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fv 6233 df-ov 7019 df-phl 20452 df-nvc 22880 df-cph 23455 |
This theorem is referenced by: ishl2 23656 csschl 23662 |
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