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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 25188 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | nlmlmod 24683 | . 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 2099 LModclmod 20832 NrmModcnlm 24577 ℂPreHilccph 25182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5303 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-xp 5680 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fv 6554 df-ov 7419 df-nlm 24583 df-cph 25184 |
This theorem is referenced by: cphclm 25205 cph2ass 25229 cphtcphnm 25246 nmparlem 25255 cphipval2 25257 4cphipval2 25258 cphipval 25259 cphsscph 25267 cmscsscms 25389 minveclem1 25440 minveclem2 25442 minveclem4 25448 minveclem6 25450 pjthlem1 25453 pjthlem2 25454 |
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