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Theorem lspfval 17746
Description: The span function for a left vector space (or a left module). (df-span 26354 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 3118 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lspval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2516 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 4020 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
7 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 lspval.s . . . . . . . 8  |-  S  =  ( LSubSp `  W )
97, 8syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
10 rabeq 3103 . . . . . . 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 4293 . . . . 5  |-  ( w  =  W  ->  |^| { t  e.  ( LSubSp `  w
)  |  s  C_  t }  =  |^| { t  e.  S  | 
s  C_  t }
)
136, 12mpteq12dv 4535 . . . 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 17745 . . . 4  |-  LSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (
LSubSp `  w )  |  s  C_  t }
) )
15 fvex 5882 . . . . . . 7  |-  ( Base `  W )  e.  _V
164, 15eqeltri 2541 . . . . . 6  |-  V  e. 
_V
1716pwex 4639 . . . . 5  |-  ~P V  e.  _V
1817mptex 6144 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  e.  _V
1913, 14, 18fvmpt 5956 . . 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 2510 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   |^|cint 4288    |-> cmpt 4515   ` cfv 5594   Basecbs 14644   LSubSpclss 17705   LSpanclspn 17744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-lsp 17745
This theorem is referenced by:  lspf  17747  lspval  17748  00lsp  17754  mrclsp  17762  lsppropd  17791
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