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Mirrors > Home > MPE Home > Th. List > clmvsdi | Structured version Visualization version GIF version |
Description: Distributive law for scalar product (left-distributivity). (lmodvsdi 20500 analog.) (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) |
clmvsdir.a | β’ + = (+gβπ) |
Ref | Expression |
---|---|
clmvsdi | β’ ((π β βMod β§ (π΄ β πΎ β§ π β π β§ π β π)) β (π΄ Β· (π + π)) = ((π΄ Β· π) + (π΄ Β· π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 24590 | . 2 β’ (π β βMod β π β LMod) | |
2 | clmvscl.v | . . 3 β’ π = (Baseβπ) | |
3 | clmvsdir.a | . . 3 β’ + = (+gβπ) | |
4 | clmvscl.f | . . 3 β’ πΉ = (Scalarβπ) | |
5 | clmvscl.s | . . 3 β’ Β· = ( Β·π βπ) | |
6 | clmvscl.k | . . 3 β’ πΎ = (BaseβπΉ) | |
7 | 2, 3, 4, 5, 6 | lmodvsdi 20500 | . 2 β’ ((π β LMod β§ (π΄ β πΎ β§ π β π β§ π β π)) β (π΄ Β· (π + π)) = ((π΄ Β· π) + (π΄ Β· π))) |
8 | 1, 7 | sylan 580 | 1 β’ ((π β βMod β§ (π΄ β πΎ β§ π β π β§ π β π)) β (π΄ Β· (π + π)) = ((π΄ Β· π) + (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 6543 (class class class)co 7411 Basecbs 17146 +gcplusg 17199 Scalarcsca 17202 Β·π cvsca 17203 LModclmod 20475 βModcclm 24585 |
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 2703 ax-nul 5306 |
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 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7414 df-lmod 20477 df-clm 24586 |
This theorem is referenced by: clmnegsubdi2 24628 clmsub4 24629 ncvspi 24680 |
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