Theorem lsatlspsn2 34279

Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 34280 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
lsatset.v	|- V = ( Base ` W )
lsatset.n	|- N = ( LSpan ` W )
lsatset.z	|- .0. = ( 0g ` W )
lsatset.a	|- A = (LSAtoms ` W )

Assertion

Ref	Expression
lsatlspsn2	|- ( ( W e. LMod /\ X e. V /\ X =/= .0. ) -> ( N ` { X } ) e. A )

Proof of Theorem lsatlspsn2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpc 1060	. . . 4 |- ( ( W e. LMod /\ X e. V /\ X =/= .0. ) -> ( X e. V /\ X =/= .0. ) )
2		eldifsn 4317	. . . 4 |- ( X e. ( V \ { .0. } ) <-> ( X e. V /\ X =/= .0. ) )
3	1, 2	sylibr 224	. . 3 |- ( ( W e. LMod /\ X e. V /\ X =/= .0. ) -> X e. ( V \ { .0. } ) )
4		eqid 2622	. . 3 |- ( N ` { X } ) = ( N ` { X } )
5		sneq 4187	. . . . . 6 |- ( v = X -> { v } = { X } )
6	5	fveq2d 6195	. . . . 5 |- ( v = X -> ( N ` { v } ) = ( N ` { X } ) )
7	6	eqeq2d 2632	. . . 4 |- ( v = X -> ( ( N ` { X } ) = ( N ` { v } ) <-> ( N ` { X } ) = ( N ` { X } ) ) )
8	7	rspcev 3309	. . 3 |- ( ( X e. ( V \ { .0. } ) /\ ( N ` { X } ) = ( N ` { X } ) ) -> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) )
9	3, 4, 8	sylancl 694	. 2 |- ( ( W e. LMod /\ X e. V /\ X =/= .0. ) -> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) )
10		lsatset.v	. . . 4 |- V = ( Base ` W )
11		lsatset.n	. . . 4 |- N = ( LSpan ` W )
12		lsatset.z	. . . 4 |- .0. = ( 0g ` W )
13		lsatset.a	. . . 4 |- A = (LSAtoms ` W )
14	10, 11, 12, 13	islsat 34278	. . 3 |- ( W e. LMod -> ( ( N ` { X } ) e. A <-> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) ) )
15	14	3ad2ant1 1082	. 2 |- ( ( W e. LMod /\ X e. V /\ X =/= .0. ) -> ( ( N ` { X } ) e. A <-> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) ) )
16	9, 15	mpbird 247	1 |- ( ( W e. LMod /\ X e. V /\ X =/= .0. ) -> ( N ` { X } ) e. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177   ` cfv 5888   Basecbs 15857   0gc0g 16100   LModclmod 18863   LSpanclspn 18971  LSAtomsclsa 34261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-lsatoms 34263

This theorem is referenced by:  lsatel  34292  lsmsat  34295  lssatomic  34298  lssats  34299  dihlsprn  36620  dihatlat  36623  dihatexv  36627  dochsatshpb  36741