Theorem lsssn0 14709

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 2239	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2239	. 2 |- ( W e. LMod -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
3		eqidd 2239	. 2 |- ( W e. LMod -> ( Base ` W ) = ( Base ` W ) )
4		eqidd 2239	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
5		eqidd 2239	. 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 2238	. . . 4 |- ( Base ` W ) = ( Base ` W )
9		lss0cl.z	. . . 4 |- .0. = ( 0g ` W )
10	8, 9	lmod0vcl 14656	. . 3 |- ( W e. LMod -> .0. e. ( Base ` W ) )
11	10	snssd 3860	. 2 |- ( W e. LMod -> { .0. } C_ ( Base ` W ) )
12		snmg 3831	. . 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 1035	. . . . . . . 8 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> a e. { .0. } )
15		elsni 3727	. . . . . . . 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 6101	. . . . . 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 2238	. . . . . . . 8 |- (Scalar ` W ) = (Scalar ` W )
19		eqid 2238	. . . . . . . 8 |- ( .s ` W ) = ( .s ` W )
20		eqid 2238	. . . . . . . 8 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
21	18, 19, 20, 9	lmodvs0 14661	. . . . . . 7 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) ) -> ( x ( .s ` W ) .0. ) = .0. )
22	21	3ad2antr1 1193	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) .0. ) = .0. )
23	17, 22	eqtrd 2271	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> ( x ( .s ` W ) a ) = .0. )
24		simpr3 1036	. . . . . 6 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. { .0. } /\ b e. { .0. } ) ) -> b e. { .0. } )
25		elsni 3727	. . . . . 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 6103	. . . 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 2238	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
29	8, 28, 9	lmod0vlid 14657	. . . . . 6 |- ( ( W e. LMod /\ .0. e. ( Base ` W ) ) -> ( .0. ( +g ` W ) .0. ) = .0. )
30	10, 29	mpdan 425	. . . . 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 2271	. . 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 2824	. . . . . . . 8 |- x e. _V
34	33	a1i 9	. . . . . . 7 |- ( W e. LMod -> x e. _V )
35		vscaslid 13519	. . . . . . . 8 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
36	35	slotex 13381	. . . . . . 7 |- ( W e. LMod -> ( .s ` W ) e. _V )
37		vex 2824	. . . . . . . 8 |- a e. _V
38	37	a1i 9	. . . . . . 7 |- ( W e. LMod -> a e. _V )
39		ovexg 6119	. . . . . . 7 |- ( ( x e. _V /\ ( .s ` W ) e. _V /\ a e. _V ) -> ( x ( .s ` W ) a ) e. _V )
40	34, 36, 38, 39	syl3anc 1278	. . . . . 6 |- ( W e. LMod -> ( x ( .s ` W ) a ) e. _V )
41		plusgslid 13468	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
42	41	slotex 13381	. . . . . 6 |- ( W e. LMod -> ( +g ` W ) e. _V )
43		vex 2824	. . . . . . 7 |- b e. _V
44	43	a1i 9	. . . . . 6 |- ( W e. LMod -> b e. _V )
45		ovexg 6119	. . . . . 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 1278	. . . . 5 |- ( W e. LMod -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. _V )
47		elsng 3724	. . . . 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 14698	1 |- ( W e. LMod -> { .0. } e. S )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   {csn 3709   ` cfv 5377  (class class class)co 6085   Basecbs 13354   +g cplusg 13433  Scalarcsca 13436   .scvsca 13437   0gc0g 13612   LModclmod 14625   LSubSpclss 14691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-plusg 13446  df-mulr 13447  df-sca 13449  df-vsca 13450  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-mgp 14220  df-ring 14304  df-lmod 14627  df-lssm 14692

This theorem is used by:  lspsn0  14761  lsp0  14762  lidl0  14828