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| Mirrors > Home > ILE Home > Th. List > lmodvscl | GIF version | ||
| Description: Closure of scalar product for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lmodvscl.v | β’ π = (Baseβπ) |
| lmodvscl.f | β’ πΉ = (Scalarβπ) |
| lmodvscl.s | β’ Β· = ( Β·π βπ) |
| lmodvscl.k | β’ πΎ = (BaseβπΉ) |
| Ref | Expression |
|---|---|
| lmodvscl | β’ ((π β LMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biid 171 | . 2 β’ (π β LMod β π β LMod) | |
| 2 | pm4.24 395 | . 2 β’ (π β πΎ β (π β πΎ β§ π β πΎ)) | |
| 3 | pm4.24 395 | . 2 β’ (π β π β (π β π β§ π β π)) | |
| 4 | lmodvscl.v | . . . . 5 β’ π = (Baseβπ) | |
| 5 | eqid 2177 | . . . . 5 β’ (+gβπ) = (+gβπ) | |
| 6 | lmodvscl.s | . . . . 5 β’ Β· = ( Β·π βπ) | |
| 7 | lmodvscl.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
| 8 | lmodvscl.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
| 9 | eqid 2177 | . . . . 5 β’ (+gβπΉ) = (+gβπΉ) | |
| 10 | eqid 2177 | . . . . 5 β’ (.rβπΉ) = (.rβπΉ) | |
| 11 | eqid 2177 | . . . . 5 β’ (1rβπΉ) = (1rβπΉ) | |
| 12 | 4, 5, 6, 7, 8, 9, 10, 11 | lmodlema 13387 | . . . 4 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ) β§ (π β π β§ π β π)) β (((π Β· π) β π β§ (π Β· (π(+gβπ)π)) = ((π Β· π)(+gβπ)(π Β· π)) β§ ((π (+gβπΉ)π ) Β· π) = ((π Β· π)(+gβπ)(π Β· π))) β§ (((π (.rβπΉ)π ) Β· π) = (π Β· (π Β· π)) β§ ((1rβπΉ) Β· π) = π))) |
| 13 | 12 | simpld 112 | . . 3 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ) β§ (π β π β§ π β π)) β ((π Β· π) β π β§ (π Β· (π(+gβπ)π)) = ((π Β· π)(+gβπ)(π Β· π)) β§ ((π (+gβπΉ)π ) Β· π) = ((π Β· π)(+gβπ)(π Β· π)))) |
| 14 | 13 | simp1d 1009 | . 2 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ) β§ (π β π β§ π β π)) β (π Β· π) β π) |
| 15 | 1, 2, 3, 14 | syl3anb 1281 | 1 β’ ((π β LMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 978 = wceq 1353 β wcel 2148 βcfv 5218 (class class class)co 5877 Basecbs 12464 +gcplusg 12538 .rcmulr 12539 Scalarcsca 12541 Β·π cvsca 12542 1rcur 13147 LModclmod 13382 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-ov 5880 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-ndx 12467 df-slot 12468 df-base 12470 df-plusg 12551 df-mulr 12552 df-sca 12554 df-vsca 12555 df-lmod 13384 |
| This theorem is referenced by: lmodscaf 13405 lmod0vs 13416 lmodvsmmulgdi 13418 lcomf 13422 lmodvneg1 13425 lmodvsneg 13426 lmodnegadd 13431 lmodsubvs 13438 lmodsubdi 13439 lmodsubdir 13440 lmodprop2d 13443 lss1 13454 lssvsubcl 13457 lssvscl 13467 lss1d 13475 |
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