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 23778 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | nlmlmod 23289 | . 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 2114 LModclmod 19636 NrmModcnlm 23192 ℂPreHilccph 23772 |
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 2795 ax-nul 5212 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fv 6365 df-ov 7161 df-nlm 23198 df-cph 23774 |
This theorem is referenced by: cphclm 23795 cph2ass 23819 cphtcphnm 23835 nmparlem 23844 cphipval2 23846 4cphipval2 23847 cphipval 23848 cphsscph 23856 cmscsscms 23978 minveclem1 24029 minveclem2 24031 minveclem4 24037 minveclem6 24039 pjthlem1 24042 pjthlem2 24043 |
Copyright terms: Public domain | W3C validator |