Theorem lmod0vs 18896

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27867 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lmod0vs.v	|- V = ( Base ` W )
lmod0vs.f	|- F = (Scalar ` W )
lmod0vs.s	|- .x. = ( .s ` W )
lmod0vs.o	|- O = ( 0g ` F )
lmod0vs.z	|- .0. = ( 0g ` W )

Assertion

Ref	Expression
lmod0vs	|- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. )

Proof of Theorem lmod0vs

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> W e. LMod )
2		lmod0vs.f	. . . . . . . 8 |- F = (Scalar ` W )
3	2	lmodring 18871	. . . . . . 7 |- ( W e. LMod -> F e. Ring )
4	3	adantr 481	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> F e. Ring )
5		eqid 2622	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
6		lmod0vs.o	. . . . . . 7 |- O = ( 0g ` F )
7	5, 6	ring0cl 18569	. . . . . 6 |- ( F e. Ring -> O e. ( Base ` F ) )
8	4, 7	syl 17	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> O e. ( Base ` F ) )
9		simpr 477	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> X e. V )
10		lmod0vs.v	. . . . . 6 |- V = ( Base ` W )
11		eqid 2622	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
12		lmod0vs.s	. . . . . 6 |- .x. = ( .s ` W )
13		eqid 2622	. . . . . 6 |- ( +g ` F ) = ( +g ` F )
14	10, 11, 2, 12, 5, 13	lmodvsdir 18887	. . . . 5 |- ( ( W e. LMod /\ ( O e. ( Base ` F ) /\ O e. ( Base ` F ) /\ X e. V ) ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) )
15	1, 8, 8, 9, 14	syl13anc 1328	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) )
16		ringgrp 18552	. . . . . . 7 |- ( F e. Ring -> F e. Grp )
17	4, 16	syl 17	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> F e. Grp )
18	5, 13, 6	grplid 17452	. . . . . 6 |- ( ( F e. Grp /\ O e. ( Base ` F ) ) -> ( O ( +g ` F ) O ) = O )
19	17, 8, 18	syl2anc 693	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( O ( +g ` F ) O ) = O )
20	19	oveq1d 6665	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( O .x. X ) )
21	15, 20	eqtr3d 2658	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) = ( O .x. X ) )
22	10, 2, 12, 5	lmodvscl 18880	. . . . 5 |- ( ( W e. LMod /\ O e. ( Base ` F ) /\ X e. V ) -> ( O .x. X ) e. V )
23	1, 8, 9, 22	syl3anc 1326	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) e. V )
24		lmod0vs.z	. . . . 5 |- .0. = ( 0g ` W )
25	10, 11, 24	lmod0vid 18895	. . . 4 |- ( ( W e. LMod /\ ( O .x. X ) e. V ) -> ( ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) = ( O .x. X ) <-> .0. = ( O .x. X ) ) )
26	23, 25	syldan 487	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) = ( O .x. X ) <-> .0. = ( O .x. X ) ) )
27	21, 26	mpbid 222	. 2 |- ( ( W e. LMod /\ X e. V ) -> .0. = ( O .x. X ) )
28	27	eqcomd 2628	1 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   Ringcrg 18547   LModclmod 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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549  df-lmod 18865

This theorem is referenced by:  lmodvs0  18897  lmodvsmmulgdi  18898  lcomfsupp  18903  lmodvneg1  18906  mptscmfsupp0  18928  lvecvs0or  19108  lssvs0or  19110  lspsneleq  19115  lspdisj  19125  lspfixed  19128  lspexch  19129  lspsolvlem  19142  lspsolv  19143  mplcoe1  19465  mplbas2  19470  ply10s0  19626  ply1scl0  19660  gsummoncoe1  19674  uvcresum  20132  frlmsslsp  20135  frlmup1  20137  frlmup2  20138  pmatcollpwscmatlem1  20594  idpm2idmp  20606  mp2pm2mplem4  20614  pm2mpmhmlem1  20623  monmat2matmon  20629  cpmidpmatlem3  20677  clm0vs  22895  plypf1  23968  lmodslmd  29757  lshpkrlem1  34397  ldual0vs  34447  lclkrlem1  36795  lcd0vs  36904  baerlem3lem1  36996  baerlem5blem1  36998  hdmap14lem2a  37159  hdmap14lem4a  37163  hdmap14lem6  37165  hgmapval0  37184  lmod0rng  41868  scmsuppss  42153  lmodvsmdi  42163  ascl0  42165  ply1mulgsumlem4  42177  lincval1  42208  lincvalsc0  42210  linc0scn0  42212  linc1  42214  ldepsprlem  42261