Theorem lmod0vs 14198

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 14172	. . . . . . 7 |- ( W e. LMod -> F e. Ring )
4	3	adantr 276	. . . . . 6 |- ( ( W e. LMod /\ X e. V ) -> F e. Ring )
5		eqid 2207	. . . . . . 7 |- ( Base ` F ) = ( Base ` F )
6		lmod0vs.o	. . . . . . 7 |- O = ( 0g ` F )
7	5, 6	ring0cl 13898	. . . . . 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 2207	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
12		lmod0vs.s	. . . . . 6 |- .x. = ( .s ` W )
13		eqid 2207	. . . . . 6 |- ( +g ` F ) = ( +g ` F )
14	10, 11, 2, 12, 5, 13	lmodvsdir 14189	. . . . 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 1252	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) )
16		ringgrp 13878	. . . . . . 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 13478	. . . . . 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 5982	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( ( O ( +g ` F ) O ) .x. X ) = ( O .x. X ) )
21	15, 20	eqtr3d 2242	. . 3 |- ( ( W e. LMod /\ X e. V ) -> ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) = ( O .x. X ) )
22	10, 2, 12, 5	lmodvscl 14182	. . . . 5 |- ( ( W e. LMod /\ O e. ( Base ` F ) /\ X e. V ) -> ( O .x. X ) e. V )
23	1, 8, 9, 22	syl3anc 1250	. . . 4 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) e. V )
24		lmod0vs.z	. . . . 5 |- .0. = ( 0g ` W )
25	10, 11, 24	lmod0vid 14197	. . . 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 2213	1 |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024  Scalarcsca 13027   .scvsca 13028   0gc0g 13203   Grpcgrp 13447   Ringcrg 13873   LModclmod 14164

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-mulr 13038  df-sca 13040  df-vsca 13041  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-ring 13875  df-lmod 14166

This theorem is referenced by:  lmodvs0  14199  lmodvsmmulgdi  14200  lmodvneg1  14207