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Theorem lspfval 15746
Description: The span function for a left vector space (or a left module). (df-span 21904 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 2809 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5541 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lspval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 3643 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
7 fveq2 5541 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 lspval.s . . . . . . . 8  |-  S  =  ( LSubSp `  W )
97, 8syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
10 rabeq 2795 . . . . . . 7  |-  ( (
LSubSp `  w )  =  S  ->  { t  e.  ( LSubSp `  w )  |  s  C_  t }  =  { t  e.  S  |  s  C_  t } )
119, 10syl 15 . . . . . 6  |-  ( w  =  W  ->  { t  e.  ( LSubSp `  w
)  |  s  C_  t }  =  {
t  e.  S  | 
s  C_  t }
)
1211inteqd 3883 . . . . 5  |-  ( w  =  W  ->  |^| { t  e.  ( LSubSp `  w
)  |  s  C_  t }  =  |^| { t  e.  S  | 
s  C_  t }
)
136, 12mpteq12dv 4114 . . . 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 15745 . . . 4  |-  LSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (
LSubSp `  w )  |  s  C_  t }
) )
15 fvex 5555 . . . . . . 7  |-  ( Base `  W )  e.  _V
164, 15eqeltri 2366 . . . . . 6  |-  V  e. 
_V
1716pwex 4209 . . . . 5  |-  ~P V  e.  _V
1817mptex 5762 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  e.  _V
1913, 14, 18fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  ( LSpan `  W )  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
202, 19syl 15 . 2  |-  ( W  e.  X  ->  ( LSpan `  W )  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
211, 20syl5eq 2340 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   |^|cint 3878    e. cmpt 4093   ` cfv 5271   Basecbs 13164   LSubSpclss 15705   LSpanclspn 15744
This theorem is referenced by:  lspf  15747  lspval  15748  00lsp  15754  mrclsp  15762  lsppropd  15791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-lsp 15745
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