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Mirrors > Home > MPE Home > Th. List > clmvscl | Structured version Visualization version GIF version |
Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20721. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.) |
Ref | Expression |
---|---|
clmvscl.v | β’ π = (Baseβπ) |
clmvscl.f | β’ πΉ = (Scalarβπ) |
clmvscl.s | β’ Β· = ( Β·π βπ) |
clmvscl.k | β’ πΎ = (BaseβπΉ) |
Ref | Expression |
---|---|
clmvscl | β’ ((π β βMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 24944 | . 2 β’ (π β βMod β π β LMod) | |
2 | clmvscl.v | . . 3 β’ π = (Baseβπ) | |
3 | clmvscl.f | . . 3 β’ πΉ = (Scalarβπ) | |
4 | clmvscl.s | . . 3 β’ Β· = ( Β·π βπ) | |
5 | clmvscl.k | . . 3 β’ πΎ = (BaseβπΉ) | |
6 | 2, 3, 4, 5 | lmodvscl 20721 | . 2 β’ ((π β LMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
7 | 1, 6 | syl3an1 1160 | 1 β’ ((π β βMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 Basecbs 17150 Scalarcsca 17206 Β·π cvsca 17207 LModclmod 20703 βModcclm 24939 |
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 6488 df-fv 6544 df-ov 7407 df-lmod 20705 df-clm 24940 |
This theorem is referenced by: clmpm1dir 24980 clmnegsubdi2 24982 clmsub4 24983 clmvsubval2 24987 clmvz 24988 nmoleub2lem3 24992 nmoleub3 24996 ncvspi 25034 |
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