Theorem lsssn0 13866

Description: The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lss0cl.z	|- .0. = ( 0g ` W )
lss0cl.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
lsssn0	|- ( W e. LMod -> { .0. } e. S )

Proof of Theorem lsssn0

Dummy variables x a b j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2194	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2194	. 2 |- ( W e. LMod -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
3		eqidd 2194	. 2 |- ( W e. LMod -> ( Base ` W ) = ( Base ` W ) )
4		eqidd 2194	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
5		eqidd 2194	. 2 |- ( W e. LMod -> ( .s ` W ) = ( .s ` W ) )
6		lss0cl.s	. . 3 |- S = ( LSubSp ` W )
7	6	a1i 9	. 2 |- ( W e. LMod -> S = ( LSubSp ` W ) )
8		eqid 2193	. . . 4 |- ( Base ` W ) = ( Base ` W )
9		lss0cl.z	. . . 4 |- .0. = ( 0g ` W )
10	8, 9	lmod0vcl 13813	. . 3 |- ( W e. LMod -> .0. e. ( Base ` W ) )
11	10	snssd 3763	. 2 |- ( W e. LMod -> { .0. } C_ ( Base ` W ) )
12		snmg 3736	. . 3 |- ( .0. e. ( Base ` W ) -> E. j j e. { .0. } )
13	10, 12	syl 14	. 2 |- ( W e. LMod -> E. j j e. { .0. } )
14		simpr2 1006	. . . . . . . 8 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> a e. { .0. } )
15		elsni 3636	. . . . . . . 8 |- ( a e. { .0. } -> a = .0. )
16	14, 15	syl 14	. . . . . . 7 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> a = .0. )
17	16	oveq2d 5934	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) a ) = ( x ( .s ` W ) .0. ) )
18		eqid 2193	. . . . . . . 8 |- (Scalar ` W ) = (Scalar ` W )
19		eqid 2193	. . . . . . . 8 |- ( .s ` W ) = ( .s ` W )
20		eqid 2193	. . . . . . . 8 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
21	18, 19, 20, 9	lmodvs0 13818	. . . . . . 7 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) ) -> ( x ( .s ` W ) .0. ) = .0. )
22	21	3ad2antr1 1164	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) .0. ) = .0. )
23	17, 22	eqtrd 2226	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) a ) = .0. )
24		simpr3 1007	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> b e. { .0. } )
25		elsni 3636	. . . . . 6 |- ( b e. { .0. } -> b = .0. )
26	24, 25	syl 14	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> b = .0. )
27	23, 26	oveq12d 5936	. . . 4 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) = ( .0. ( +g ` W ) .0. ) )
28		eqid 2193	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
29	8, 28, 9	lmod0vlid 13814	. . . . . 6 |- ( ( W e. LMod /\ .0. e. ( Base ` W ) ) -> ( .0. ( +g ` W ) .0. ) = .0. )
30	10, 29	mpdan 421	. . . . 5 |- ( W e. LMod -> ( .0. ( +g ` W ) .0. ) = .0. )
31	30	adantr 276	. . . 4 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( .0. ( +g ` W ) .0. ) = .0. )
32	27, 31	eqtrd 2226	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) = .0. )
33		vex 2763	. . . . . . . 8 |- x e. _V
34	33	a1i 9	. . . . . . 7 |- ( W e. LMod -> x e. _V )
35		vscaslid 12780	. . . . . . . 8 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
36	35	slotex 12645	. . . . . . 7 |- ( W e. LMod -> ( .s ` W ) e. _V )
37		vex 2763	. . . . . . . 8 |- a e. _V
38	37	a1i 9	. . . . . . 7 |- ( W e. LMod -> a e. _V )
39		ovexg 5952	. . . . . . 7 |- ( ( x e. _V /\ ( .s ` W ) e. _V /\ a e. _V ) -> ( x ( .s ` W ) a ) e. _V )
40	34, 36, 38, 39	syl3anc 1249	. . . . . 6 |- ( W e. LMod -> ( x ( .s ` W ) a ) e. _V )
41		plusgslid 12730	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
42	41	slotex 12645	. . . . . 6 |- ( W e. LMod -> ( +g ` W ) e. _V )
43		vex 2763	. . . . . . 7 |- b e. _V
44	43	a1i 9	. . . . . 6 |- ( W e. LMod -> b e. _V )
45		ovexg 5952	. . . . . 6 |- ( ( ( x ( .s ` W ) a ) e. _V /\ ( +g ` W ) e. _V /\ b e. _V ) -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. _V )
46	40, 42, 44, 45	syl3anc 1249	. . . . 5 |- ( W e. LMod -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. _V )
47		elsng 3633	. . . . 5 |- ( ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. _V -> ( ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. { .0. } <-> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) = .0. ) )
48	46, 47	syl 14	. . . 4 |- ( W e. LMod -> ( ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. { .0. } <-> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) = .0. ) )
49	48	adantr 276	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. { .0. } <-> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) = .0. ) )
50	32, 49	mpbird 167	. 2 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. { .0. } )
51		id 19	. 2 |- ( W e. LMod -> W e. LMod )
52	1, 2, 3, 4, 5, 7, 11, 13, 50, 51	islssmd 13855	1 |- ( W e. LMod -> { .0. } e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   {csn 3618   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695  Scalarcsca 12698   .scvsca 12699   0gc0g 12867   LModclmod 13783   LSubSpclss 13848

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-in1 615  ax-in2 616  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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-mgp 13417  df-ring 13494  df-lmod 13785  df-lssm 13849

This theorem is referenced by:  lspsn0  13918  lsp0  13919  lidl0  13985