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Metamath Proof Explorer

Theorem lsatset

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)

Ref Expression
Hypotheses lsatset.v 
|- V = ( Base ` W )
lsatset.n 
|- N = ( LSpan ` W )
lsatset.z 
|- .0. = ( 0g ` W )
lsatset.a 
|- A = ( LSAtoms ` W )
Assertion lsatset 
|- ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )

Proof

Step Hyp Ref Expression
1 lsatset.v 
 |-  V = ( Base ` W )
2 lsatset.n 
 |-  N = ( LSpan ` W )
3 lsatset.z 
 |-  .0. = ( 0g ` W )
4 lsatset.a 
 |-  A = ( LSAtoms ` W )
5 elex 
 |-  ( W e. X -> W e. _V )
6 fveq2 
 |-  ( w = W -> ( Base ` w ) = ( Base ` W ) )
7 6 1 eqtr4di 
 |-  ( w = W -> ( Base ` w ) = V )
8 fveq2 
 |-  ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
9 8 3 eqtr4di 
 |-  ( w = W -> ( 0g ` w ) = .0. )
10 9 sneqd 
 |-  ( w = W -> { ( 0g ` w ) } = { .0. } )
11 7 10 difeq12d 
 |-  ( w = W -> ( ( Base ` w ) \ { ( 0g ` w ) } ) = ( V \ { .0. } ) )
12 fveq2 
 |-  ( w = W -> ( LSpan ` w ) = ( LSpan ` W ) )
13 12 2 eqtr4di 
 |-  ( w = W -> ( LSpan ` w ) = N )
14 13 fveq1d 
 |-  ( w = W -> ( ( LSpan ` w ) ` { v } ) = ( N ` { v } ) )
15 11 14 mpteq12dv 
 |-  ( w = W -> ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
16 15 rneqd 
 |-  ( w = W -> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
17 df-lsatoms 
 |-  LSAtoms = ( w e. _V |-> ran ( v e. ( ( Base ` w ) \ { ( 0g ` w ) } ) |-> ( ( LSpan ` w ) ` { v } ) ) )
18 2 fvexi 
 |-  N e. _V
19 18 rnex 
 |-  ran N e. _V
20 snex 
 |-  { (/) } e. _V
21 19 20 unex 
 |-  ( ran N u. { (/) } ) e. _V
22 eqid 
 |-  ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) = ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) )
23 fvrn0 
 |-  ( N ` { v } ) e. ( ran N u. { (/) } )
24 23 a1i 
 |-  ( v e. ( V \ { .0. } ) -> ( N ` { v } ) e. ( ran N u. { (/) } ) )
25 22 24 fmpti 
 |-  ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } )
26 frn 
 |-  ( ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) : ( V \ { .0. } ) --> ( ran N u. { (/) } ) -> ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) C_ ( ran N u. { (/) } ) )
27 25 26 ax-mp 
 |-  ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) C_ ( ran N u. { (/) } )
28 21 27 ssexi 
 |-  ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) e. _V
29 16 17 28 fvmpt 
 |-  ( W e. _V -> ( LSAtoms ` W ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
30 5 29 syl 
 |-  ( W e. X -> ( LSAtoms ` W ) = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )
31 4 30 eqtrid 
 |-  ( W e. X -> A = ran ( v e. ( V \ { .0. } ) |-> ( N ` { v } ) ) )