Theorem lsatset 34277

Description: The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)

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
lsatset	|- ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )

Distinct variable groups:   v, N   v, V   v, W   v, .0.   v, X

Allowed substitution hint:   A( v)

Proof of Theorem lsatset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsatset.a	. 2 |- A = (LSAtoms ` W )
2		elex 3212	. . 3 |- ( W e. X -> W e. _V )
3		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lsatset.v	. . . . . . . 8 |- V = ( Base ` W )
5	3, 4	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
6		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
7		lsatset.z	. . . . . . . . 9 |- .0. = ( 0g ` W )
8	6, 7	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( 0g ` w ) = .0. )
9	8	sneqd 4189	. . . . . . 7 |- ( w = W -> { ( 0g ` w ) } = { .0. } )
10	5, 9	difeq12d 3729	. . . . . 6 |- ( w = W -> ( ( Base ` w ) \ { ( 0g ` w ) } ) = ( V \ { .0. } ) )
11		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( LSpan ` w ) = ( LSpan ` W ) )
12		lsatset.n	. . . . . . . 8 |- N = ( LSpan ` W )
13	11, 12	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( LSpan ` w ) = N )
14	13	fveq1d 6193	. . . . . 6 |- ( w = W -> ( ( LSpan ` w ) ` { v } ) = ( N ` { v } ) )
15	10, 14	mpteq12dv 4733	. . . . 5 |- ( w = W -> ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
16	15	rneqd 5353	. . . 4 |- ( w = W -> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
17		df-lsatoms 34263	. . . 4 |- LSAtoms = ( w e. _V |-> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) )
18		fvex 6201	. . . . . . . 8 |- ( LSpan ` W ) e. _V
19	12, 18	eqeltri 2697	. . . . . . 7 |- N e. _V
20	19	rnex 7100	. . . . . 6 |- ran N e. _V
21		p0ex 4853	. . . . . 6 |- { (/) } e. _V
22	20, 21	unex 6956	. . . . 5 |- ( ran N u. { (/) } ) e. _V
23		eqid 2622	. . . . . . 7 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) )
24		fvrn0 6216	. . . . . . . 8 |- ( N ` { v } ) e. ( ran N u. { (/) } )
25	24	a1i 11	. . . . . . 7 |- ( v e. ( V \ { .0. } ) -> ( N ` { v } ) e. ( ran N u. { (/) } ) )
26	23, 25	fmpti 6383	. . . . . 6 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } )
27		frn 6053	. . . . . 6 |- ( ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } ) -> ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) C_ ( ran N u. { (/) } ) )
28	26, 27	ax-mp 5	. . . . 5 |- ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) C_ ( ran N u. { (/) } )
29	22, 28	ssexi 4803	. . . 4 |- ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) e. _V
30	16, 17, 29	fvmpt 6282	. . 3 |- ( W e. _V -> (LSAtoms ` W ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
31	2, 30	syl 17	. 2 |- ( W e. X -> (LSAtoms ` W ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
32	1, 31	syl5eq 2668	1 |- ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888   Basecbs 15857   0gc0g 16100   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:  islsat  34278  lsatlss  34283