Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23775 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
2 | phllvec 20773 | . 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 2114 LVecclvec 19874 PreHilcphl 20768 ℂPreHilccph 23770 |
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 2793 ax-nul 5210 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fv 6363 df-ov 7159 df-phl 20770 df-cph 23772 |
This theorem is referenced by: cphnvc 23780 cphsubrg 23784 cphreccl 23785 cphqss 23792 hlprlem 23970 ishl2 23973 |
Copyright terms: Public domain | W3C validator |