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Mirrors > Home > MPE Home > Th. List > lmodvsdir | Structured version Visualization version Unicode version |
Description: Distributive law for scalar product (right-distributivity). (ax-hvdistr1 27865 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
lmodvsdir.v | |
lmodvsdir.a | |
lmodvsdir.f | Scalar |
lmodvsdir.s | |
lmodvsdir.k | |
lmodvsdir.p |
Ref | Expression |
---|---|
lmodvsdir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvsdir.v | . . . . . . . 8 | |
2 | lmodvsdir.a | . . . . . . . 8 | |
3 | lmodvsdir.s | . . . . . . . 8 | |
4 | lmodvsdir.f | . . . . . . . 8 Scalar | |
5 | lmodvsdir.k | . . . . . . . 8 | |
6 | lmodvsdir.p | . . . . . . . 8 | |
7 | eqid 2622 | . . . . . . . 8 | |
8 | eqid 2622 | . . . . . . . 8 | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmodlema 18868 | . . . . . . 7 |
10 | 9 | simpld 475 | . . . . . 6 |
11 | 10 | simp3d 1075 | . . . . 5 |
12 | 11 | 3expa 1265 | . . . 4 |
13 | 12 | anabsan2 863 | . . 3 |
14 | 13 | exp42 639 | . 2 |
15 | 14 | 3imp2 1282 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmulr 15942 Scalarcsca 15944 cvsca 15945 cur 18501 clmod 18863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-lmod 18865 |
This theorem is referenced by: lmod0vs 18896 lmodvsmmulgdi 18898 lmodvneg1 18906 lmodcom 18909 lmodsubdir 18921 islss3 18959 lss1d 18963 prdslmodd 18969 lspsolvlem 19142 asclghm 19338 frlmup1 20137 scmataddcl 20322 scmatghm 20339 pm2mpghm 20621 clmvsdir 22891 cvsi 22930 lshpkrlem4 34400 baerlem3lem1 36996 baerlem5blem1 36998 hgmapadd 37186 mendlmod 37763 lmodvsmdi 42163 lincsum 42218 ldepsprlem 42261 |
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