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Mirrors > Home > MPE Home > Th. List > phllmod | Structured version Visualization version GIF version |
Description: A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phllmod | ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phllvec 19793 | . 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
2 | lveclmod 18927 | . 2 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 LModclmod 18686 LVecclvec 18923 PreHilcphl 19788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-iota 5768 df-fv 5812 df-ov 6552 df-lvec 18924 df-phl 19790 |
This theorem is referenced by: iporthcom 19799 ip0l 19800 ip0r 19801 ipdir 19803 ipdi 19804 ip2di 19805 ipsubdir 19806 ipsubdi 19807 ip2subdi 19808 ipass 19809 ipassr 19810 ip2eq 19817 phssip 19822 ocvlss 19835 ocvin 19837 ocvlsp 19839 ocvz 19841 ocv1 19842 lsmcss 19855 pjdm2 19874 pjff 19875 pjf2 19877 pjfo 19878 ocvpj 19880 obselocv 19891 obslbs 19893 tchclm 22839 ipcau2 22841 tchcphlem1 22842 tchcphlem2 22843 tchcph 22844 pjth 23018 |
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