Theorem lsatset 32943

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 3079	. . 3 |- ( W e. X -> W e. _V )
3		fveq2 5791	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lsatset.v	. . . . . . . 8 |- V = ( Base ` W )
5	3, 4	syl6eqr 2510	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
6		fveq2 5791	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
7		lsatset.z	. . . . . . . . 9 |- .0. = ( 0g ` W )
8	6, 7	syl6eqr 2510	. . . . . . . 8 |- ( w = W -> ( 0g ` w ) = .0. )
9	8	sneqd 3989	. . . . . . 7 |- ( w = W -> { ( 0g ` w ) } = { .0. } )
10	5, 9	difeq12d 3575	. . . . . 6 |- ( w = W -> ( ( Base ` w ) \ { ( 0g ` w ) } ) = ( V \ { .0. } ) )
11		fveq2 5791	. . . . . . . 8 |- ( w = W -> ( LSpan ` w ) = ( LSpan ` W ) )
12		lsatset.n	. . . . . . . 8 |- N = ( LSpan ` W )
13	11, 12	syl6eqr 2510	. . . . . . 7 |- ( w = W -> ( LSpan ` w ) = N )
14	13	fveq1d 5793	. . . . . 6 |- ( w = W -> ( ( LSpan ` w ) ` { v } ) = ( N ` { v } ) )
15	10, 14	mpteq12dv 4470	. . . . 5 |- ( w = W -> ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
16	15	rneqd 5167	. . . 4 |- ( w = W -> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
17		df-lsatoms 32929	. . . 4 |- LSAtoms = ( w e. _V |-> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) )
18		fvex 5801	. . . . . . . 8 |- ( LSpan ` W ) e. _V
19	12, 18	eqeltri 2535	. . . . . . 7 |- N e. _V
20	19	rnex 6614	. . . . . 6 |- ran N e. _V
21		p0ex 4579	. . . . . 6 |- { (/) } e. _V
22	20, 21	unex 6480	. . . . 5 |- ( ran N u. { (/) } ) e. _V
23		eqid 2451	. . . . . . 7 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) )
24		fvrn0 5813	. . . . . . . 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 5967	. . . . . 6 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } )
27		frn 5665	. . . . . 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 4537	. . . 4 |- ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) e. _V
30	16, 17, 29	fvmpt 5875	. . 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 2504	1 |- ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   _Vcvv 3070   \ cdif 3425   u. cun 3426   C_ wss 3428   (/)c0 3737   {csn 3977   |-> cmpt 4450   ran crn 4941   -->wf 5514   ` cfv 5518   Basecbs 14278   0gc0g 14482   LSpanclspn 17160  LSAtomsclsa 32927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-lsatoms 32929

This theorem is referenced by:  islsat  32944  lsatlss  32949