Theorem lsssn0 14647

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 2235	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2235	. 2 |- ( W e. LMod -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
3		eqidd 2235	. 2 |- ( W e. LMod -> ( Base ` W ) = ( Base ` W ) )
4		eqidd 2235	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
5		eqidd 2235	. 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 2234	. . . 4 |- ( Base ` W ) = ( Base ` W )
9		lss0cl.z	. . . 4 |- .0. = ( 0g ` W )
10	8, 9	lmod0vcl 14594	. . 3 |- ( W e. LMod -> .0. e. ( Base ` W ) )
11	10	snssd 3844	. 2 |- ( W e. LMod -> { .0. } C_ ( Base ` W ) )
12		snmg 3815	. . 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 1031	. . . . . . . 8 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> a e. { .0. } )
15		elsni 3712	. . . . . . . 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 6074	. . . . . 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 2234	. . . . . . . 8 |- (Scalar ` W ) = (Scalar ` W )
19		eqid 2234	. . . . . . . 8 |- ( .s ` W ) = ( .s ` W )
20		eqid 2234	. . . . . . . 8 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
21	18, 19, 20, 9	lmodvs0 14599	. . . . . . 7 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) ) -> ( x ( .s ` W ) .0. ) = .0. )
22	21	3ad2antr1 1189	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) .0. ) = .0. )
23	17, 22	eqtrd 2267	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) a ) = .0. )
24		simpr3 1032	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> b e. { .0. } )
25		elsni 3712	. . . . . 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 6076	. . . 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 2234	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
29	8, 28, 9	lmod0vlid 14595	. . . . . 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 2267	. . 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 2818	. . . . . . . 8 |- x e. _V
34	33	a1i 9	. . . . . . 7 |- ( W e. LMod -> x e. _V )
35		vscaslid 13463	. . . . . . . 8 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
36	35	slotex 13326	. . . . . . 7 |- ( W e. LMod -> ( .s ` W ) e. _V )
37		vex 2818	. . . . . . . 8 |- a e. _V
38	37	a1i 9	. . . . . . 7 |- ( W e. LMod -> a e. _V )
39		ovexg 6092	. . . . . . 7 |- ( ( x e. _V /\ ( .s ` W ) e. _V /\ a e. _V ) -> ( x ( .s ` W ) a ) e. _V )
40	34, 36, 38, 39	syl3anc 1274	. . . . . 6 |- ( W e. LMod -> ( x ( .s ` W ) a ) e. _V )
41		plusgslid 13412	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
42	41	slotex 13326	. . . . . 6 |- ( W e. LMod -> ( +g ` W ) e. _V )
43		vex 2818	. . . . . . 7 |- b e. _V
44	43	a1i 9	. . . . . 6 |- ( W e. LMod -> b e. _V )
45		ovexg 6092	. . . . . 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 1274	. . . . 5 |- ( W e. LMod -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. _V )
47		elsng 3709	. . . . 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 14636	1 |- ( W e. LMod -> { .0. } e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   {csn 3694   ` cfv 5357  (class class class)co 6058   Basecbs 13299   +g cplusg 13377  Scalarcsca 13380   .scvsca 13381   0gc0g 13556   LModclmod 14564   LSubSpclss 14629

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-plusg 13390  df-mulr 13391  df-sca 13393  df-vsca 13394  df-0g 13558  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-grp 13761  df-mgp 14163  df-ring 14244  df-lmod 14566  df-lssm 14630

This theorem is referenced by:  lspsn0  14699  lsp0  14700  lidl0  14766