Theorem lmod0vs 13598

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 13572	. . . . . . 7 |- ( W e. LMod -> F e. Ring )
4	3	adantr 276	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> F e. Ring )
5		eqid 2189	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
6		lmod0vs.o	. . . . . . 7 |- O = ( 0g ` F )
7	5, 6	ring0cl 13336	. . . . . 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 2189	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
12		lmod0vs.s	. . . . . 6 |- .x. = ( .s ` W )
13		eqid 2189	. . . . . 6 |- ( +g ` F ) = ( +g ` F )
14	10, 11, 2, 12, 5, 13	lmodvsdir 13589	. . . . 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 1251	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) )
16		ringgrp 13316	. . . . . . 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 12941	. . . . . 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 5906	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( O .x. X ) )
21	15, 20	eqtr3d 2224	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) = ( O .x. X ) )
22	10, 2, 12, 5	lmodvscl 13582	. . . . 5 |- ( ( W e. LMod /\ O e. ( Base ` F ) /\ X e. V ) -> ( O .x. X ) e. V )
23	1, 8, 9, 22	syl3anc 1249	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) e. V )
24		lmod0vs.z	. . . . 5 |- .0. = ( 0g ` W )
25	10, 11, 24	lmod0vid 13597	. . . 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 2195	1 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   ` cfv 5231  (class class class)co 5891   Basecbs 12480   +g cplusg 12555  Scalarcsca 12558   .scvsca 12559   0gc0g 12727   Grpcgrp 12911   Ringcrg 13311   LModclmod 13564

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-5 8999  df-6 9000  df-ndx 12483  df-slot 12484  df-base 12486  df-plusg 12568  df-mulr 12569  df-sca 12571  df-vsca 12572  df-0g 12729  df-mgm 12798  df-sgrp 12831  df-mnd 12844  df-grp 12914  df-ring 13313  df-lmod 13566

This theorem is referenced by:  lmodvs0  13599  lmodvsmmulgdi  13600  lmodvneg1  13607