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Theorem lspfval 16054
Description: The span function for a left vector space (or a left module). (df-span 22816 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 2966 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5731 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lspval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2488 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 3806 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
7 fveq2 5731 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 lspval.s . . . . . . . 8  |-  S  =  ( LSubSp `  W )
97, 8syl6eqr 2488 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
10 rabeq 2952 . . . . . . 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 4057 . . . . 5  |-  ( w  =  W  ->  |^| { t  e.  ( LSubSp `  w
)  |  s  C_  t }  =  |^| { t  e.  S  | 
s  C_  t }
)
136, 12mpteq12dv 4290 . . . 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 16053 . . . 4  |-  LSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (
LSubSp `  w )  |  s  C_  t }
) )
15 fvex 5745 . . . . . . 7  |-  ( Base `  W )  e.  _V
164, 15eqeltri 2508 . . . . . 6  |-  V  e. 
_V
1716pwex 4385 . . . . 5  |-  ~P V  e.  _V
1817mptex 5969 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  e.  _V
1913, 14, 18fvmpt 5809 . . 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 2482 1  |-  ( W  e.  X  ->  N  =  ( s  e. 
~P V  |->  |^| { t  e.  S  |  s 
C_  t } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   |^|cint 4052    e. cmpt 4269   ` cfv 5457   Basecbs 13474   LSubSpclss 16013   LSpanclspn 16052
This theorem is referenced by:  lspf  16055  lspval  16056  00lsp  16062  mrclsp  16070  lsppropd  16099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-lsp 16053
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