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Mirrors > Home > MPE Home > Th. List > cphlvec | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
cphlvec | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphphl 22971 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
2 | phllvec 19974 | . 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 LVecclvec 19102 PreHilcphl 19969 ℂPreHilccph 22966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ov 6653 df-phl 19971 df-cph 22968 |
This theorem is referenced by: cphnvc 22976 cphsubrg 22980 cphreccl 22981 cphqss 22988 hlprlem 23163 ishl2 23166 |
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