Theorem lsssn0 13936

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 2197	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2197	. 2 |- ( W e. LMod -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
3		eqidd 2197	. 2 |- ( W e. LMod -> ( Base ` W ) = ( Base ` W ) )
4		eqidd 2197	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
5		eqidd 2197	. 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 2196	. . . 4 |- ( Base ` W ) = ( Base ` W )
9		lss0cl.z	. . . 4 |- .0. = ( 0g ` W )
10	8, 9	lmod0vcl 13883	. . 3 |- ( W e. LMod -> .0. e. ( Base ` W ) )
11	10	snssd 3768	. 2 |- ( W e. LMod -> { .0. } C_ ( Base ` W ) )
12		snmg 3741	. . 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 3641	. . . . . . . 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 5939	. . . . . 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 2196	. . . . . . . 8 |- (Scalar ` W ) = (Scalar ` W )
19		eqid 2196	. . . . . . . 8 |- ( .s ` W ) = ( .s ` W )
20		eqid 2196	. . . . . . . 8 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
21	18, 19, 20, 9	lmodvs0 13888	. . . . . . 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 2229	. . . . 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 3641	. . . . . 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 5941	. . . 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 2196	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
29	8, 28, 9	lmod0vlid 13884	. . . . . 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 2229	. . 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 2766	. . . . . . . 8 |- x e. _V
34	33	a1i 9	. . . . . . 7 |- ( W e. LMod -> x e. _V )
35		vscaslid 12850	. . . . . . . 8 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
36	35	slotex 12715	. . . . . . 7 |- ( W e. LMod -> ( .s ` W ) e. _V )
37		vex 2766	. . . . . . . 8 |- a e. _V
38	37	a1i 9	. . . . . . 7 |- ( W e. LMod -> a e. _V )
39		ovexg 5957	. . . . . . 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 12800	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
42	41	slotex 12715	. . . . . 6 |- ( W e. LMod -> ( +g ` W ) e. _V )
43		vex 2766	. . . . . . 7 |- b e. _V
44	43	a1i 9	. . . . . 6 |- ( W e. LMod -> b e. _V )
45		ovexg 5957	. . . . . 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 3638	. . . . 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 13925	1 |- ( W e. LMod -> { .0. } e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   {csn 3623   ` cfv 5259  (class class class)co 5923   Basecbs 12688   +g cplusg 12765  Scalarcsca 12768   .scvsca 12769   0gc0g 12937   LModclmod 13853   LSubSpclss 13918

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-i2m1 7986  ax-0lt1 7987  ax-0id 7989  ax-rnegex 7990  ax-pre-ltirr 7993  ax-pre-ltadd 7997

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-pnf 8065  df-mnf 8066  df-ltxr 8068  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-5 9054  df-6 9055  df-ndx 12691  df-slot 12692  df-base 12694  df-sets 12695  df-plusg 12778  df-mulr 12779  df-sca 12781  df-vsca 12782  df-0g 12939  df-mgm 13009  df-sgrp 13055  df-mnd 13068  df-grp 13145  df-mgp 13487  df-ring 13564  df-lmod 13855  df-lssm 13919

This theorem is referenced by:  lspsn0  13988  lsp0  13989  lidl0  14055