![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cphlmod | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
cphlmod | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphnlm 23192 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | nlmlmod 22703 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2139 LModclmod 19085 NrmModcnlm 22606 ℂPreHilccph 23186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-nul 4941 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-xp 5272 df-cnv 5274 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fv 6057 df-ov 6817 df-nlm 22612 df-cph 23188 |
This theorem is referenced by: cphclm 23209 cph2ass 23233 cphtchnm 23249 nmparlem 23258 cphipval2 23260 4cphipval2 23261 cphipval 23262 minveclem1 23415 minveclem2 23417 minveclem4 23423 minveclem6 23425 pjthlem1 23428 pjthlem2 23429 |
Copyright terms: Public domain | W3C validator |