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Mirrors > Home > MPE Home > Th. List > lmodvsass | Structured version Visualization version GIF version |
Description: Associative law for scalar product. (ax-hvmulass 29949 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
lmodvsass.v | β’ π = (Baseβπ) |
lmodvsass.f | β’ πΉ = (Scalarβπ) |
lmodvsass.s | β’ Β· = ( Β·π βπ) |
lmodvsass.k | β’ πΎ = (BaseβπΉ) |
lmodvsass.t | β’ Γ = (.rβπΉ) |
Ref | Expression |
---|---|
lmodvsass | β’ ((π β LMod β§ (π β πΎ β§ π β πΎ β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsass.v | . . . . . . 7 β’ π = (Baseβπ) | |
2 | eqid 2736 | . . . . . . 7 β’ (+gβπ) = (+gβπ) | |
3 | lmodvsass.s | . . . . . . 7 β’ Β· = ( Β·π βπ) | |
4 | lmodvsass.f | . . . . . . 7 β’ πΉ = (Scalarβπ) | |
5 | lmodvsass.k | . . . . . . 7 β’ πΎ = (BaseβπΉ) | |
6 | eqid 2736 | . . . . . . 7 β’ (+gβπΉ) = (+gβπΉ) | |
7 | lmodvsass.t | . . . . . . 7 β’ Γ = (.rβπΉ) | |
8 | eqid 2736 | . . . . . . 7 β’ (1rβπΉ) = (1rβπΉ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 20327 | . . . . . 6 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 Γ π ) Β· π) = (π Β· (π Β· π)) β§ ((1rβπΉ) Β· π) = π))) |
10 | 9 | simprld 770 | . . . . 5 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ) β§ (π β π β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
11 | 10 | 3expa 1118 | . . . 4 β’ (((π β LMod β§ (π β πΎ β§ π β πΎ)) β§ (π β π β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
12 | 11 | anabsan2 672 | . . 3 β’ (((π β LMod β§ (π β πΎ β§ π β πΎ)) β§ π β π) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
13 | 12 | exp42 436 | . 2 β’ (π β LMod β (π β πΎ β (π β πΎ β (π β π β ((π Γ π ) Β· π) = (π Β· (π Β· π)))))) |
14 | 13 | 3imp2 1349 | 1 β’ ((π β LMod β§ (π β πΎ β§ π β πΎ β§ π β π)) β ((π Γ π ) Β· π) = (π Β· (π Β· π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6496 (class class class)co 7357 Basecbs 17083 +gcplusg 17133 .rcmulr 17134 Scalarcsca 17136 Β·π cvsca 17137 1rcur 19913 LModclmod 20322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5263 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-iota 6448 df-fv 6504 df-ov 7360 df-lmod 20324 |
This theorem is referenced by: lmodvs0 20356 lmodvsneg 20366 lmodsubvs 20378 lmodsubdi 20379 lmodsubdir 20380 islss3 20420 lss1d 20424 prdslmodd 20430 lmodvsinv 20497 lmhmvsca 20506 lvecvs0or 20569 lssvs0or 20571 lvecinv 20574 lspsnvs 20575 lspfixed 20589 lspsolvlem 20603 lspsolv 20604 frlmup1 21204 assa2ass 21269 ascldimul 21291 assamulgscmlem2 21303 mplmon2mul 21477 smatvscl 21873 matinv 22026 clmvsass 24452 cvsi 24493 imaslmod 32145 lshpkrlem4 37575 lcdvsass 40070 baerlem3lem1 40170 hgmapmul 40358 prjspertr 40929 prjspner1 40950 mendlmod 41506 lincscm 46501 ldepsprlem 46543 lincresunit3lem3 46545 lincresunit3lem1 46550 |
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