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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 21554 | . 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
2 | lveclmod 20984 | . 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 2099 LModclmod 20736 LVecclvec 20980 PreHilcphl 21549 |
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 2699 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-iota 6494 df-fv 6550 df-ov 7417 df-lvec 20981 df-phl 21551 |
This theorem is referenced by: iporthcom 21560 ip0l 21561 ip0r 21562 ipdir 21564 ipdi 21565 ip2di 21566 ipsubdir 21567 ipsubdi 21568 ip2subdi 21569 ipass 21570 ipassr 21571 ip2eq 21578 phssip 21583 phlssphl 21584 ocvlss 21597 ocvin 21599 ocvlsp 21601 ocvz 21603 ocv1 21604 lsmcss 21617 pjdm2 21638 pjff 21639 pjf2 21641 pjfo 21642 ocvpj 21644 obselocv 21655 obslbs 21657 phclm 25153 ipcau2 25155 tcphcphlem1 25156 tcphcphlem2 25157 tcphcph 25158 pjth 25360 |
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