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Theorem aspval 16068
Description: Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
aspval.a  |-  A  =  (AlgSpan `  W )
aspval.v  |-  V  =  ( Base `  W
)
aspval.l  |-  L  =  ( LSubSp `  W )
Assertion
Ref Expression
aspval  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
Distinct variable groups:    t, L    t, S    t, V    t, W
Allowed substitution hint:    A( t)

Proof of Theorem aspval
Dummy variables  s  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aspval.a . . . . 5  |-  A  =  (AlgSpan `  W )
2 fveq2 5525 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 aspval.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2333 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  V )
54pweqd 3630 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
6 fveq2 5525 . . . . . . . . . 10  |-  ( w  =  W  ->  (SubRing `  w )  =  (SubRing `  W ) )
7 fveq2 5525 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 aspval.l . . . . . . . . . . 11  |-  L  =  ( LSubSp `  W )
97, 8syl6eqr 2333 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
106, 9ineq12d 3371 . . . . . . . . 9  |-  ( w  =  W  ->  (
(SubRing `  w )  i^i  ( LSubSp `  w )
)  =  ( (SubRing `  W )  i^i  L
) )
11 rabeq 2782 . . . . . . . . 9  |-  ( ( (SubRing `  w )  i^i  ( LSubSp `  w )
)  =  ( (SubRing `  W )  i^i  L
)  ->  { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }  =  { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )
1210, 11syl 15 . . . . . . . 8  |-  ( w  =  W  ->  { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }  =  { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )
1312inteqd 3867 . . . . . . 7  |-  ( w  =  W  ->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } )
145, 13mpteq12dv 4098 . . . . . 6  |-  ( w  =  W  ->  (
s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  ( (SubRing `  w )  i^i  ( LSubSp `  w )
)  |  s  C_  t } )  =  ( s  e.  ~P V  |-> 
|^| { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } ) )
15 df-asp 16054 . . . . . 6  |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
16 fvex 5539 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
173, 16eqeltri 2353 . . . . . . . 8  |-  V  e. 
_V
1817pwex 4193 . . . . . . 7  |-  ~P V  e.  _V
1918mptex 5746 . . . . . 6  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )  e.  _V
2014, 15, 19fvmpt 5602 . . . . 5  |-  ( W  e. AssAlg  ->  (AlgSpan `  W )  =  ( s  e. 
~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) )
211, 20syl5eq 2327 . . . 4  |-  ( W  e. AssAlg  ->  A  =  ( s  e.  ~P V  |-> 
|^| { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } ) )
2221fveq1d 5527 . . 3  |-  ( W  e. AssAlg  ->  ( A `  S )  =  ( ( s  e.  ~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) `  S ) )
2322adantr 451 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  ( ( s  e.  ~P V  |->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } ) `
 S ) )
24 simpr 447 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  V )
2517elpw2 4175 . . . 4  |-  ( S  e.  ~P V  <->  S  C_  V
)
2624, 25sylibr 203 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  e.  ~P V )
27 assarng 16061 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  Ring )
283subrgid 15547 . . . . . . 7  |-  ( W  e.  Ring  ->  V  e.  (SubRing `  W )
)
2927, 28syl 15 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  (SubRing `  W ) )
30 assalmod 16060 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
313, 8lss1 15696 . . . . . . 7  |-  ( W  e.  LMod  ->  V  e.  L )
3230, 31syl 15 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  L
)
33 elin 3358 . . . . . 6  |-  ( V  e.  ( (SubRing `  W
)  i^i  L )  <->  ( V  e.  (SubRing `  W
)  /\  V  e.  L ) )
3429, 32, 33sylanbrc 645 . . . . 5  |-  ( W  e. AssAlg  ->  V  e.  ( (SubRing `  W )  i^i  L ) )
35 sseq2 3200 . . . . . 6  |-  ( t  =  V  ->  ( S  C_  t  <->  S  C_  V
) )
3635rspcev 2884 . . . . 5  |-  ( ( V  e.  ( (SubRing `  W )  i^i  L
)  /\  S  C_  V
)  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
3734, 36sylan 457 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
38 intexrab 4170 . . . 4  |-  ( E. t  e.  ( (SubRing `  W )  i^i  L
) S  C_  t  <->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  S  C_  t }  e.  _V )
3937, 38sylib 188 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t }  e.  _V )
40 sseq1 3199 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  t  <->  S  C_  t
) )
4140rabbidv 2780 . . . . 5  |-  ( s  =  S  ->  { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t }  =  { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
4241inteqd 3867 . . . 4  |-  ( s  =  S  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t }  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
43 eqid 2283 . . . 4  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )  =  ( s  e. 
~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } )
4442, 43fvmptg 5600 . . 3  |-  ( ( S  e.  ~P V  /\  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t }  e.  _V )  -> 
( ( s  e. 
~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
4526, 39, 44syl2anc 642 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  (
( s  e.  ~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
4623, 45eqtrd 2315 1  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  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    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   |^|cint 3862    e. cmpt 4077   ` cfv 5255   Basecbs 13148   Ringcrg 15337  SubRingcsubrg 15541   LModclmod 15627   LSubSpclss 15689  AssAlgcasa 16050  AlgSpancasp 16051
This theorem is referenced by:  asplss  16069  aspid  16070  aspsubrg  16071  aspss  16072  aspssid  16073  aspval2  16086
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  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-assa 16053  df-asp 16054
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