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Theorem lspval 15732
Description: The span of a set of vectors (in a left module). (spanval 21912 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 15730 . . . 4  |-  ( W  e.  LMod  ->  N  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
54fveq1d 5527 . . 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 5539 . . . . . 6  |-  ( Base `  W )  e.  _V
91, 8eqeltri 2353 . . . . 5  |-  V  e. 
_V
109elpw2 4175 . . . 4  |-  ( U  e.  ~P V  <->  U  C_  V
)
117, 10sylibr 203 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  e.  ~P V )
121, 2lss1 15696 . . . . 5  |-  ( W  e.  LMod  ->  V  e.  S )
13 sseq2 3200 . . . . . 6  |-  ( t  =  V  ->  ( U  C_  t  <->  U  C_  V
) )
1413rspcev 2884 . . . . 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 4170 . . . 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 3199 . . . . . 6  |-  ( s  =  U  ->  (
s  C_  t  <->  U  C_  t
) )
1918rabbidv 2780 . . . . 5  |-  ( s  =  U  ->  { t  e.  S  |  s 
C_  t }  =  { t  e.  S  |  U  C_  t } )
2019inteqd 3867 . . . 4  |-  ( s  =  U  ->  |^| { t  e.  S  |  s 
C_  t }  =  |^| { t  e.  S  |  U  C_  t } )
21 eqid 2283 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  =  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )
2220, 21fvmptg 5600 . . 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 2315 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 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   |^|cint 3862    e. cmpt 4077   ` cfv 5255   Basecbs 13148   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728
This theorem is referenced by:  lspid  15739  lspss  15741  lspssid  15742  dochspss  31568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690  df-lsp 15729
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