Theorem lmod0vs 13411

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (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 109	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> W e. LMod )
2		lmod0vs.f	. . . . . . . 8 |- F = (Scalar ` W )
3	2	lmodring 13385	. . . . . . 7 |- ( W e. LMod -> F e. Ring )
4	3	adantr 276	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> F e. Ring )
5		eqid 2177	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
6		lmod0vs.o	. . . . . . 7 |- O = ( 0g ` F )
7	5, 6	ring0cl 13204	. . . . . 6 |- ( F e. Ring -> O e. ( Base ` F ) )
8	4, 7	syl 14	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> O e. ( Base ` F ) )
9		simpr 110	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> X e. V )
10		lmod0vs.v	. . . . . 6 |- V = ( Base ` W )
11		eqid 2177	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
12		lmod0vs.s	. . . . . 6 |- .x. = ( .s ` W )
13		eqid 2177	. . . . . 6 |- ( +g ` F ) = ( +g ` F )
14	10, 11, 2, 12, 5, 13	lmodvsdir 13402	. . . . 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 1240	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) )
16		ringgrp 13184	. . . . . . 7 |- ( F e. Ring -> F e. Grp )
17	4, 16	syl 14	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> F e. Grp )
18	5, 13, 6	grplid 12906	. . . . . 6 |- ( ( F e. Grp /\ O e. ( Base ` F ) ) -> ( O ( +g ` F ) O ) = O )
19	17, 8, 18	syl2anc 411	. . . . 5 |- ( ( W e. LMod /\ X e. V ) -> ( O ( +g ` F ) O ) = O )
20	19	oveq1d 5890	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( O .x. X ) )
21	15, 20	eqtr3d 2212	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) = ( O .x. X ) )
22	10, 2, 12, 5	lmodvscl 13395	. . . . 5 |- ( ( W e. LMod /\ O e. ( Base ` F ) /\ X e. V ) -> ( O .x. X ) e. V )
23	1, 8, 9, 22	syl3anc 1238	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) e. V )
24		lmod0vs.z	. . . . 5 |- .0. = ( 0g ` W )
25	10, 11, 24	lmod0vid 13410	. . . 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 282	. . 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 147	. 2 |- ( ( W e. LMod /\ X e. V ) -> .0. = ( O .x. X ) )
28	27	eqcomd 2183	1 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536  Scalarcsca 12539   .scvsca 12540   0gc0g 12705   Grpcgrp 12877   Ringcrg 13179   LModclmod 13377

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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

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-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-riota 5831  df-ov 5878  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-5 8981  df-6 8982  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-mulr 12550  df-sca 12552  df-vsca 12553  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-ring 13181  df-lmod 13379

This theorem is referenced by:  lmodvs0  13412  lmodvsmmulgdi  13413  lmodvneg1  13420