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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 20372 | . 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
2 | lveclmod 19501 | . 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 2106 LModclmod 19255 LVecclvec 19497 PreHilcphl 20367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-iota 6099 df-fv 6143 df-ov 6925 df-lvec 19498 df-phl 20369 |
This theorem is referenced by: iporthcom 20378 ip0l 20379 ip0r 20380 ipdir 20382 ipdi 20383 ip2di 20384 ipsubdir 20385 ipsubdi 20386 ip2subdi 20387 ipass 20388 ipassr 20389 ip2eq 20396 phssip 20401 phlssphl 20402 ocvlss 20415 ocvin 20417 ocvlsp 20419 ocvz 20421 ocv1 20422 lsmcss 20435 pjdm2 20454 pjff 20455 pjf2 20457 pjfo 20458 ocvpj 20460 obselocv 20471 obslbs 20473 phclm 23438 ipcau2 23440 tcphcphlem1 23441 tcphcphlem2 23442 tcphcph 23443 pjth 23645 |
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