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Theorem lspval 17943
Description: The span of a set of vectors (in a left module). (spanval 26678 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 17941 . . . 4  |-  ( W  e.  LMod  ->  N  =  ( s  e.  ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
54fveq1d 5853 . . 3  |-  ( W  e.  LMod  ->  ( N `
 U )  =  ( ( s  e. 
~P V  |->  |^| { t  e.  S  |  s 
C_  t } ) `
 U ) )
65adantr 465 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  ( ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } ) `  U ) )
7 simpr 461 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  C_  V )
8 fvex 5861 . . . . . 6  |-  ( Base `  W )  e.  _V
91, 8eqeltri 2488 . . . . 5  |-  V  e. 
_V
109elpw2 4560 . . . 4  |-  ( U  e.  ~P V  <->  U  C_  V
)
117, 10sylibr 214 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  e.  ~P V )
121, 2lss1 17907 . . . . 5  |-  ( W  e.  LMod  ->  V  e.  S )
13 sseq2 3466 . . . . . 6  |-  ( t  =  V  ->  ( U  C_  t  <->  U  C_  V
) )
1413rspcev 3162 . . . . 5  |-  ( ( V  e.  S  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t )
1512, 14sylan 471 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  E. t  e.  S  U  C_  t
)
16 intexrab 4555 . . . 4  |-  ( E. t  e.  S  U  C_  t  <->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
1715, 16sylib 198 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  |^| { t  e.  S  |  U  C_  t }  e.  _V )
18 sseq1 3465 . . . . . 6  |-  ( s  =  U  ->  (
s  C_  t  <->  U  C_  t
) )
1918rabbidv 3053 . . . . 5  |-  ( s  =  U  ->  { t  e.  S  |  s 
C_  t }  =  { t  e.  S  |  U  C_  t } )
2019inteqd 4234 . . . 4  |-  ( s  =  U  ->  |^| { t  e.  S  |  s 
C_  t }  =  |^| { t  e.  S  |  U  C_  t } )
21 eqid 2404 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )  =  ( s  e.  ~P V  |->  |^|
{ t  e.  S  |  s  C_  t } )
2220, 21fvmptg 5932 . . 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 661 . 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 2445 1  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844   E.wrex 2757   {crab 2760   _Vcvv 3061    C_ wss 3416   ~Pcpw 3957   |^|cint 4229    |-> cmpt 4455   ` cfv 5571   Basecbs 14843   LModclmod 17834   LSubSpclss 17900   LSpanclspn 17939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-grp 16383  df-lmod 17836  df-lss 17901  df-lsp 17940
This theorem is referenced by:  lspid  17950  lspss  17952  lspssid  17953  dochspss  34411  lcosslsp  38563
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