MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspfval Structured version   Unicode version

Theorem lspfval 17059
Description: The span function for a left vector space (or a left module). (df-span 24717 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 2986 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5696 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lspval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 3870 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
7 fveq2 5696 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 lspval.s . . . . . . . 8  |-  S  =  ( LSubSp `  W )
97, 8syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
10 rabeq 2971 . . . . . . 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 4138 . . . . 5  |-  ( w  =  W  ->  |^| { t  e.  ( LSubSp `  w
)  |  s  C_  t }  =  |^| { t  e.  S  | 
s  C_  t }
)
136, 12mpteq12dv 4375 . . . 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 17058 . . . 4  |-  LSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (
LSubSp `  w )  |  s  C_  t }
) )
15 fvex 5706 . . . . . . 7  |-  ( Base `  W )  e.  _V
164, 15eqeltri 2513 . . . . . 6  |-  V  e. 
_V
1716pwex 4480 . . . . 5  |-  ~P V  e.  _V
1817mptex 5953 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  e.  _V
1913, 14, 18fvmpt 5779 . . 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 2487 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 1369    e. wcel 1756   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   |^|cint 4133    e. cmpt 4355   ` cfv 5423   Basecbs 14179   LSubSpclss 17018   LSpanclspn 17057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-lsp 17058
This theorem is referenced by:  lspf  17060  lspval  17061  00lsp  17067  mrclsp  17075  lsppropd  17104
  Copyright terms: Public domain W3C validator