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Mirrors > Home > MPE Home > Th. List > lmodvscld | Structured version Visualization version GIF version |
Description: Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.) |
Ref | Expression |
---|---|
lmodvscld.v | β’ π = (Baseβπ) |
lmodvscld.f | β’ πΉ = (Scalarβπ) |
lmodvscld.s | β’ Β· = ( Β·π βπ) |
lmodvscld.k | β’ πΎ = (BaseβπΉ) |
lmodvscld.w | β’ (π β π β LMod) |
lmodvscld.r | β’ (π β π β πΎ) |
lmodvscld.x | β’ (π β π β π) |
Ref | Expression |
---|---|
lmodvscld | β’ (π β (π Β· π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvscld.w | . 2 β’ (π β π β LMod) | |
2 | lmodvscld.r | . 2 β’ (π β π β πΎ) | |
3 | lmodvscld.x | . 2 β’ (π β π β π) | |
4 | lmodvscld.v | . . 3 β’ π = (Baseβπ) | |
5 | lmodvscld.f | . . 3 β’ πΉ = (Scalarβπ) | |
6 | lmodvscld.s | . . 3 β’ Β· = ( Β·π βπ) | |
7 | lmodvscld.k | . . 3 β’ πΎ = (BaseβπΉ) | |
8 | 4, 5, 6, 7 | lmodvscl 20724 | . 2 β’ ((π β LMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
9 | 1, 2, 3, 8 | syl3anc 1368 | 1 β’ (π β (π Β· π) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 Basecbs 17153 Scalarcsca 17209 Β·π cvsca 17210 LModclmod 20706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 df-lmod 20708 |
This theorem is referenced by: q1pvsca 33179 r1pvsca 33180 r1plmhm 33185 ply1degltdimlem 33225 algextdeglem8 33301 frlmsnic 41667 selvvvval 41714 prjspvs 41930 prjspeclsp 41932 |
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