Theorem lsatset 33662

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 3115	. . 3 |- ( W e. X -> W e. _V )
3		fveq2 5857	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lsatset.v	. . . . . . . 8 |- V = ( Base ` W )
5	3, 4	syl6eqr 2519	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
6		fveq2 5857	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
7		lsatset.z	. . . . . . . . 9 |- .0. = ( 0g ` W )
8	6, 7	syl6eqr 2519	. . . . . . . 8 |- ( w = W -> ( 0g ` w ) = .0. )
9	8	sneqd 4032	. . . . . . 7 |- ( w = W -> { ( 0g ` w ) } = { .0. } )
10	5, 9	difeq12d 3616	. . . . . 6 |- ( w = W -> ( ( Base ` w ) \ { ( 0g ` w ) } ) = ( V \ { .0. } ) )
11		fveq2 5857	. . . . . . . 8 |- ( w = W -> ( LSpan ` w ) = ( LSpan ` W ) )
12		lsatset.n	. . . . . . . 8 |- N = ( LSpan ` W )
13	11, 12	syl6eqr 2519	. . . . . . 7 |- ( w = W -> ( LSpan ` w ) = N )
14	13	fveq1d 5859	. . . . . 6 |- ( w = W -> ( ( LSpan ` w ) ` { v } ) = ( N ` { v } ) )
15	10, 14	mpteq12dv 4518	. . . . 5 |- ( w = W -> ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
16	15	rneqd 5221	. . . 4 |- ( w = W -> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
17		df-lsatoms 33648	. . . 4 |- LSAtoms = ( w e. _V |-> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) )
18		fvex 5867	. . . . . . . 8 |- ( LSpan ` W ) e. _V
19	12, 18	eqeltri 2544	. . . . . . 7 |- N e. _V
20	19	rnex 6708	. . . . . 6 |- ran N e. _V
21		p0ex 4627	. . . . . 6 |- { (/) } e. _V
22	20, 21	unex 6573	. . . . 5 |- ( ran N u. { (/) } ) e. _V
23		eqid 2460	. . . . . . 7 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) )
24		fvrn0 5879	. . . . . . . 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 6035	. . . . . 6 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } )
27		frn 5728	. . . . . 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 4585	. . . 4 |- ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) e. _V
30	16, 17, 29	fvmpt 5941	. . 3 |- ( W e. _V -> (LSAtoms ` W ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
31	2, 30	syl 16	. 2 |- ( W e. X -> (LSAtoms ` W ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
32	1, 31	syl5eq 2513	1 |- ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )

Syntax hints:   -> wi 4   = wceq 1374   e. wcel 1762   _Vcvv 3106   \ cdif 3466   u. cun 3467   C_ wss 3469   (/)c0 3778   {csn 4020   |-> cmpt 4498   ran crn 4993   -->wf 5575   ` cfv 5579   Basecbs 14479   0gc0g 14684   LSpanclspn 17393  LSAtomsclsa 33646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-lsatoms 33648

This theorem is referenced by:  islsat  33663  lsatlss  33668