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Theorem lspfval 17399
Description: The span function for a left vector space (or a left module). (df-span 25900 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lspval.v  |-  V  =  ( Base `  W
)
lspval.s  |-  S  =  ( LSubSp `  W )
lspval.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
lspfval  |-  ( W  e.  X  ->  N  =  ( s  e. 
~P V  |->  |^| { t  e.  S  |  s 
C_  t } ) )
Distinct variable groups:    t, s, S    V, s, t    W, s
Allowed substitution hints:    N( t, s)    W( t)    X( t, s)

Proof of Theorem lspfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 lspval.n . 2  |-  N  =  ( LSpan `  W )
2 elex 3122 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5864 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lspval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 4015 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
7 fveq2 5864 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 lspval.s . . . . . . . 8  |-  S  =  ( LSubSp `  W )
97, 8syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
10 rabeq 3107 . . . . . . 7  |-  ( (
LSubSp `  w )  =  S  ->  { t  e.  ( LSubSp `  w )  |  s  C_  t }  =  { t  e.  S  |  s  C_  t } )
119, 10syl 16 . . . . . 6  |-  ( w  =  W  ->  { t  e.  ( LSubSp `  w
)  |  s  C_  t }  =  {
t  e.  S  | 
s  C_  t }
)
1211inteqd 4287 . . . . 5  |-  ( w  =  W  ->  |^| { t  e.  ( LSubSp `  w
)  |  s  C_  t }  =  |^| { t  e.  S  | 
s  C_  t }
)
136, 12mpteq12dv 4525 . . . 4  |-  ( w  =  W  ->  (
s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (
LSubSp `  w )  |  s  C_  t }
)  =  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } ) )
14 df-lsp 17398 . . . 4  |-  LSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (
LSubSp `  w )  |  s  C_  t }
) )
15 fvex 5874 . . . . . . 7  |-  ( Base `  W )  e.  _V
164, 15eqeltri 2551 . . . . . 6  |-  V  e. 
_V
1716pwex 4630 . . . . 5  |-  ~P V  e.  _V
1817mptex 6129 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  e.  _V
1913, 14, 18fvmpt 5948 . . 3  |-  ( W  e.  _V  ->  ( LSpan `  W )  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
202, 19syl 16 . 2  |-  ( W  e.  X  ->  ( LSpan `  W )  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
211, 20syl5eq 2520 1  |-  ( W  e.  X  ->  N  =  ( s  e. 
~P V  |->  |^| { t  e.  S  |  s 
C_  t } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   |^|cint 4282    |-> cmpt 4505   ` cfv 5586   Basecbs 14483   LSubSpclss 17358   LSpanclspn 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-lsp 17398
This theorem is referenced by:  lspf  17400  lspval  17401  00lsp  17407  mrclsp  17415  lsppropd  17444
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