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

Theorem lspval 15831
Description: The span of a set of vectors (in a left module). (spanval 22026 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
lspval  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
Distinct variable groups:    t, S    t, U    t, V
Allowed substitution hints:    N( t)    W( t)

Proof of Theorem lspval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 lspval.v . . . . 5  |-  V  =  ( Base `  W
)
2 lspval.s . . . . 5  |-  S  =  ( LSubSp `  W )
3 lspval.n . . . . 5  |-  N  =  ( LSpan `  W )
41, 2, 3lspfval 15829 . . . 4  |-  ( W  e.  LMod  ->  N  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
54fveq1d 5610 . . 3  |-  ( W  e.  LMod  ->  ( N `
 U )  =  ( ( s  e. 
~P V  |->  |^| { t  e.  S  |  s 
C_  t } ) `
 U ) )
65adantr 451 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  ( ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } ) `  U ) )
7 simpr 447 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  C_  V )
8 fvex 5622 . . . . . 6  |-  ( Base `  W )  e.  _V
91, 8eqeltri 2428 . . . . 5  |-  V  e. 
_V
109elpw2 4256 . . . 4  |-  ( U  e.  ~P V  <->  U  C_  V
)
117, 10sylibr 203 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  e.  ~P V )
121, 2lss1 15795 . . . . 5  |-  ( W  e.  LMod  ->  V  e.  S )
13 sseq2 3276 . . . . . 6  |-  ( t  =  V  ->  ( U  C_  t  <->  U  C_  V
) )
1413rspcev 2960 . . . . 5  |-  ( ( V  e.  S  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t )
1512, 14sylan 457 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t
)
16 intexrab 4251 . . . 4  |-  ( E. t  e.  S  U  C_  t  <->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
1715, 16sylib 188 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
18 sseq1 3275 . . . . . 6  |-  ( s  =  U  ->  (
s  C_  t  <->  U  C_  t
) )
1918rabbidv 2856 . . . . 5  |-  ( s  =  U  ->  { t  e.  S  |  s 
C_  t }  =  { t  e.  S  |  U  C_  t } )
2019inteqd 3948 . . . 4  |-  ( s  =  U  ->  |^| { t  e.  S  |  s 
C_  t }  =  |^| { t  e.  S  |  U  C_  t } )
21 eqid 2358 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  =  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )
2220, 21fvmptg 5683 . . 3  |-  ( ( U  e.  ~P V  /\  |^| { t  e.  S  |  U  C_  t }  e.  _V )  ->  ( ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } ) `  U )  =  |^| { t  e.  S  |  U  C_  t } )
2311, 17, 22syl2anc 642 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  (
( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) `  U
)  =  |^| { t  e.  S  |  U  C_  t } )
246, 23eqtrd 2390 1  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620   {crab 2623   _Vcvv 2864    C_ wss 3228   ~Pcpw 3701   |^|cint 3943    e. cmpt 4158   ` cfv 5337   Basecbs 13245   LModclmod 15726   LSubSpclss 15788   LSpanclspn 15827
This theorem is referenced by:  lspid  15838  lspss  15840  lspssid  15841  dochspss  31637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-riota 6391  df-0g 13503  df-mnd 14466  df-grp 14588  df-lmod 15728  df-lss 15789  df-lsp 15828
  Copyright terms: Public domain W3C validator