Theorem lsatset 32475

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 2976	. . 3 |- ( W e. X -> W e. _V )
3		fveq2 5686	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lsatset.v	. . . . . . . 8 |- V = ( Base ` W )
5	3, 4	syl6eqr 2488	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
6		fveq2 5686	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
7		lsatset.z	. . . . . . . . 9 |- .0. = ( 0g ` W )
8	6, 7	syl6eqr 2488	. . . . . . . 8 |- ( w = W -> ( 0g ` w ) = .0. )
9	8	sneqd 3884	. . . . . . 7 |- ( w = W -> { ( 0g ` w ) } = { .0. } )
10	5, 9	difeq12d 3470	. . . . . 6 |- ( w = W -> ( ( Base ` w ) \ { ( 0g ` w ) } ) = ( V \ { .0. } ) )
11		fveq2 5686	. . . . . . . 8 |- ( w = W -> ( LSpan ` w ) = ( LSpan ` W ) )
12		lsatset.n	. . . . . . . 8 |- N = ( LSpan ` W )
13	11, 12	syl6eqr 2488	. . . . . . 7 |- ( w = W -> ( LSpan ` w ) = N )
14	13	fveq1d 5688	. . . . . 6 |- ( w = W -> ( ( LSpan ` w ) ` { v } ) = ( N ` { v } ) )
15	10, 14	mpteq12dv 4365	. . . . 5 |- ( w = W -> ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
16	15	rneqd 5062	. . . 4 |- ( w = W -> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
17		df-lsatoms 32461	. . . 4 |- LSAtoms = ( w e. _V |-> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) )
18		fvex 5696	. . . . . . . 8 |- ( LSpan ` W ) e. _V
19	12, 18	eqeltri 2508	. . . . . . 7 |- N e. _V
20	19	rnex 6507	. . . . . 6 |- ran N e. _V
21		p0ex 4474	. . . . . 6 |- { (/) } e. _V
22	20, 21	unex 6373	. . . . 5 |- ( ran N u. { (/) } ) e. _V
23		eqid 2438	. . . . . . 7 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) )
24		fvrn0 5707	. . . . . . . 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 5861	. . . . . 6 |- ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } )
27		frn 5560	. . . . . 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 4432	. . . 4 |- ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) e. _V
30	16, 17, 29	fvmpt 5769	. . 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 2482	1 |- ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   _Vcvv 2967   \ cdif 3320   u. cun 3321   C_ wss 3323   (/)c0 3632   {csn 3872   e. cmpt 4345   ran crn 4836   -->wf 5409   ` cfv 5413   Basecbs 14166   0gc0g 14370   LSpanclspn 17029  LSAtomsclsa 32459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-lsatoms 32461

This theorem is referenced by:  islsat  32476  lsatlss  32481