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Theorem lspval 16010
Description: The span of a set of vectors (in a left module). (spanval 22792 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 16008 . . . 4  |-  ( W  e.  LMod  ->  N  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
54fveq1d 5693 . . 3  |-  ( W  e.  LMod  ->  ( N `
 U )  =  ( ( s  e. 
~P V  |->  |^| { t  e.  S  |  s 
C_  t } ) `
 U ) )
65adantr 452 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  ( ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } ) `  U ) )
7 simpr 448 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  C_  V )
8 fvex 5705 . . . . . 6  |-  ( Base `  W )  e.  _V
91, 8eqeltri 2478 . . . . 5  |-  V  e. 
_V
109elpw2 4328 . . . 4  |-  ( U  e.  ~P V  <->  U  C_  V
)
117, 10sylibr 204 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  e.  ~P V )
121, 2lss1 15974 . . . . 5  |-  ( W  e.  LMod  ->  V  e.  S )
13 sseq2 3334 . . . . . 6  |-  ( t  =  V  ->  ( U  C_  t  <->  U  C_  V
) )
1413rspcev 3016 . . . . 5  |-  ( ( V  e.  S  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t )
1512, 14sylan 458 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t
)
16 intexrab 4323 . . . 4  |-  ( E. t  e.  S  U  C_  t  <->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
1715, 16sylib 189 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
18 sseq1 3333 . . . . . 6  |-  ( s  =  U  ->  (
s  C_  t  <->  U  C_  t
) )
1918rabbidv 2912 . . . . 5  |-  ( s  =  U  ->  { t  e.  S  |  s 
C_  t }  =  { t  e.  S  |  U  C_  t } )
2019inteqd 4019 . . . 4  |-  ( s  =  U  ->  |^| { t  e.  S  |  s 
C_  t }  =  |^| { t  e.  S  |  U  C_  t } )
21 eqid 2408 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  =  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )
2220, 21fvmptg 5767 . . 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 643 . 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 2440 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 359    = wceq 1649    e. wcel 1721   E.wrex 2671   {crab 2674   _Vcvv 2920    C_ wss 3284   ~Pcpw 3763   |^|cint 4014    e. cmpt 4230   ` cfv 5417   Basecbs 13428   LModclmod 15909   LSubSpclss 15967   LSpanclspn 16006
This theorem is referenced by:  lspid  16017  lspss  16019  lspssid  16020  dochspss  31865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-riota 6512  df-0g 13686  df-mnd 14649  df-grp 14771  df-lmod 15911  df-lss 15968  df-lsp 16007
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