Theorem lsatset 34858

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 3118	. . 3 |- ( W e. X -> W e. _V )
3		fveq2 5872	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lsatset.v	. . . . . . . 8 |- V = ( Base ` W )
5	3, 4	syl6eqr 2516	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
6		fveq2 5872	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
7		lsatset.z	. . . . . . . . 9 |- .0. = ( 0g ` W )
8	6, 7	syl6eqr 2516	. . . . . . . 8 |- ( w = W -> ( 0g ` w ) = .0. )
9	8	sneqd 4044	. . . . . . 7 |- ( w = W -> { ( 0g ` w ) } = { .0. } )
10	5, 9	difeq12d 3619	. . . . . 6 |- ( w = W -> ( ( Base ` w ) \ { ( 0g ` w ) } ) = ( V \ { .0. } ) )
11		fveq2 5872	. . . . . . . 8 |- ( w = W -> ( LSpan ` w ) = ( LSpan ` W ) )
12		lsatset.n	. . . . . . . 8 |- N = ( LSpan ` W )
13	11, 12	syl6eqr 2516	. . . . . . 7 |- ( w = W -> ( LSpan ` w ) = N )
14	13	fveq1d 5874	. . . . . 6 |- ( w = W -> ( ( LSpan ` w ) ` { v } ) = ( N ` { v } ) )
15	10, 14	mpteq12dv 4535	. . . . 5 |- ( w = W -> ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
16	15	rneqd 5240	. . . 4 |- ( w = W -> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
17		df-lsatoms 34844	. . . 4 |- LSAtoms = ( w e. _V |-> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) )
18		fvex 5882	. . . . . . . 8 |- ( LSpan ` W ) e. _V
19	12, 18	eqeltri 2541	. . . . . . 7 |- N e. _V
20	19	rnex 6733	. . . . . 6 |- ran N e. _V
21		p0ex 4643	. . . . . 6 |- { (/) } e. _V
22	20, 21	unex 6597	. . . . 5 |- ( ran N u. { (/) } ) e. _V
23		eqid 2457	. . . . . . 7 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) )
24		fvrn0 5894	. . . . . . . 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 6055	. . . . . 6 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } )
27		frn 5743	. . . . . 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 4601	. . . 4 |- ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) e. _V
30	16, 17, 29	fvmpt 5956	. . 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 2510	1 |- ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 1819   _Vcvv 3109   \ cdif 3468   u. cun 3469   C_ wss 3471   (/)c0 3793   {csn 4032   |-> cmpt 4515   ran crn 5009   -->wf 5590   ` cfv 5594   Basecbs 14644   0gc0g 14857   LSpanclspn 17744  LSAtomsclsa 34842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-lsatoms 34844

This theorem is referenced by:  islsat  34859  lsatlss  34864