Theorem lsssn0 14408

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 2231	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2231	. 2 |- ( W e. LMod -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
3		eqidd 2231	. 2 |- ( W e. LMod -> ( Base ` W ) = ( Base ` W ) )
4		eqidd 2231	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
5		eqidd 2231	. 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 2230	. . . 4 |- ( Base ` W ) = ( Base ` W )
9		lss0cl.z	. . . 4 |- .0. = ( 0g ` W )
10	8, 9	lmod0vcl 14355	. . 3 |- ( W e. LMod -> .0. e. ( Base ` W ) )
11	10	snssd 3819	. 2 |- ( W e. LMod -> { .0. } C_ ( Base ` W ) )
12		snmg 3791	. . 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 1030	. . . . . . . 8 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> a e. { .0. } )
15		elsni 3688	. . . . . . . 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 6039	. . . . . 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 2230	. . . . . . . 8 |- (Scalar ` W ) = (Scalar ` W )
19		eqid 2230	. . . . . . . 8 |- ( .s ` W ) = ( .s ` W )
20		eqid 2230	. . . . . . . 8 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
21	18, 19, 20, 9	lmodvs0 14360	. . . . . . 7 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) ) -> ( x ( .s ` W ) .0. ) = .0. )
22	21	3ad2antr1 1188	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) .0. ) = .0. )
23	17, 22	eqtrd 2263	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) a ) = .0. )
24		simpr3 1031	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> b e. { .0. } )
25		elsni 3688	. . . . . 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 6041	. . . 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 2230	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
29	8, 28, 9	lmod0vlid 14356	. . . . . 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 2263	. . 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 2804	. . . . . . . 8 |- x e. _V
34	33	a1i 9	. . . . . . 7 |- ( W e. LMod -> x e. _V )
35		vscaslid 13269	. . . . . . . 8 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
36	35	slotex 13132	. . . . . . 7 |- ( W e. LMod -> ( .s ` W ) e. _V )
37		vex 2804	. . . . . . . 8 |- a e. _V
38	37	a1i 9	. . . . . . 7 |- ( W e. LMod -> a e. _V )
39		ovexg 6057	. . . . . . 7 |- ( ( x e. _V /\ ( .s ` W ) e. _V /\ a e. _V ) -> ( x ( .s ` W ) a ) e. _V )
40	34, 36, 38, 39	syl3anc 1273	. . . . . 6 |- ( W e. LMod -> ( x ( .s ` W ) a ) e. _V )
41		plusgslid 13218	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
42	41	slotex 13132	. . . . . 6 |- ( W e. LMod -> ( +g ` W ) e. _V )
43		vex 2804	. . . . . . 7 |- b e. _V
44	43	a1i 9	. . . . . 6 |- ( W e. LMod -> b e. _V )
45		ovexg 6057	. . . . . 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 1273	. . . . 5 |- ( W e. LMod -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. _V )
47		elsng 3685	. . . . 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 14397	1 |- ( W e. LMod -> { .0. } e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2201   _Vcvv 2801   {csn 3670   ` cfv 5328  (class class class)co 6023   Basecbs 13105   +g cplusg 13183  Scalarcsca 13186   .scvsca 13187   0gc0g 13362   LModclmod 14325   LSubSpclss 14390

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 619  ax-in2 620  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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-plusg 13196  df-mulr 13197  df-sca 13199  df-vsca 13200  df-0g 13364  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-grp 13609  df-mgp 13958  df-ring 14035  df-lmod 14327  df-lssm 14391

This theorem is referenced by:  lspsn0  14460  lsp0  14461  lidl0  14527