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Theorem lspval 16056
Description: The span of a set of vectors (in a left module). (spanval 22840 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 16054 . . . 4  |-  ( W  e.  LMod  ->  N  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
54fveq1d 5733 . . 3  |-  ( W  e.  LMod  ->  ( N `
 U )  =  ( ( s  e. 
~P V  |->  |^| { t  e.  S  |  s 
C_  t } ) `
 U ) )
65adantr 453 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  ( ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } ) `  U ) )
7 simpr 449 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  C_  V )
8 fvex 5745 . . . . . 6  |-  ( Base `  W )  e.  _V
91, 8eqeltri 2508 . . . . 5  |-  V  e. 
_V
109elpw2 4367 . . . 4  |-  ( U  e.  ~P V  <->  U  C_  V
)
117, 10sylibr 205 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  e.  ~P V )
121, 2lss1 16020 . . . . 5  |-  ( W  e.  LMod  ->  V  e.  S )
13 sseq2 3372 . . . . . 6  |-  ( t  =  V  ->  ( U  C_  t  <->  U  C_  V
) )
1413rspcev 3054 . . . . 5  |-  ( ( V  e.  S  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t )
1512, 14sylan 459 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t
)
16 intexrab 4362 . . . 4  |-  ( E. t  e.  S  U  C_  t  <->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
1715, 16sylib 190 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
18 sseq1 3371 . . . . . 6  |-  ( s  =  U  ->  (
s  C_  t  <->  U  C_  t
) )
1918rabbidv 2950 . . . . 5  |-  ( s  =  U  ->  { t  e.  S  |  s 
C_  t }  =  { t  e.  S  |  U  C_  t } )
2019inteqd 4057 . . . 4  |-  ( s  =  U  ->  |^| { t  e.  S  |  s 
C_  t }  =  |^| { t  e.  S  |  U  C_  t } )
21 eqid 2438 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  =  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )
2220, 21fvmptg 5807 . . 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 644 . 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 2470 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 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   |^|cint 4052    e. cmpt 4269   ` cfv 5457   Basecbs 13474   LModclmod 15955   LSubSpclss 16013   LSpanclspn 16052
This theorem is referenced by:  lspid  16063  lspss  16065  lspssid  16066  dochspss  32250
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-13 1728  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-rmo 2715  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-ov 6087  df-riota 6552  df-0g 13732  df-mnd 14695  df-grp 14817  df-lmod 15957  df-lss 16014  df-lsp 16053
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