Theorem lmod0vs 14334

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 14308	. . . . . . 7 |- ( W e. LMod -> F e. Ring )
4	3	adantr 276	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> F e. Ring )
5		eqid 2231	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
6		lmod0vs.o	. . . . . . 7 |- O = ( 0g ` F )
7	5, 6	ring0cl 14033	. . . . . 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 2231	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
12		lmod0vs.s	. . . . . 6 |- .x. = ( .s ` W )
13		eqid 2231	. . . . . 6 |- ( +g ` F ) = ( +g ` F )
14	10, 11, 2, 12, 5, 13	lmodvsdir 14325	. . . . 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 1275	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) )
16		ringgrp 14013	. . . . . . 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 13613	. . . . . 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 6032	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( O .x. X ) )
21	15, 20	eqtr3d 2266	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) = ( O .x. X ) )
22	10, 2, 12, 5	lmodvscl 14318	. . . . 5 |- ( ( W e. LMod /\ O e. ( Base ` F ) /\ X e. V ) -> ( O .x. X ) e. V )
23	1, 8, 9, 22	syl3anc 1273	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) e. V )
24		lmod0vs.z	. . . . 5 |- .0. = ( 0g ` W )
25	10, 11, 24	lmod0vid 14333	. . . 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 2237	1 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159  Scalarcsca 13162   .scvsca 13163   0gc0g 13338   Grpcgrp 13582   Ringcrg 14008   LModclmod 14300

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mulr 13173  df-sca 13175  df-vsca 13176  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-ring 14010  df-lmod 14302

This theorem is referenced by:  lmodvs0  14335  lmodvsmmulgdi  14336  lmodvneg1  14343