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Theorem aspval 17764
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 5865 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 aspval.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2526 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  V )
54pweqd 4015 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
6 fveq2 5865 . . . . . . . . . 10  |-  ( w  =  W  ->  (SubRing `  w )  =  (SubRing `  W ) )
7 fveq2 5865 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 aspval.l . . . . . . . . . . 11  |-  L  =  ( LSubSp `  W )
97, 8syl6eqr 2526 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
106, 9ineq12d 3701 . . . . . . . . 9  |-  ( w  =  W  ->  (
(SubRing `  w )  i^i  ( LSubSp `  w )
)  =  ( (SubRing `  W )  i^i  L
) )
11 rabeq 3107 . . . . . . . . 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 16 . . . . . . . 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 4287 . . . . . . 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 4525 . . . . . 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 17749 . . . . . 6  |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
16 fvex 5875 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
173, 16eqeltri 2551 . . . . . . . 8  |-  V  e. 
_V
1817pwex 4630 . . . . . . 7  |-  ~P V  e.  _V
1918mptex 6130 . . . . . 6  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )  e.  _V
2014, 15, 19fvmpt 5949 . . . . 5  |-  ( W  e. AssAlg  ->  (AlgSpan `  W )  =  ( s  e. 
~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) )
211, 20syl5eq 2520 . . . 4  |-  ( W  e. AssAlg  ->  A  =  ( s  e.  ~P V  |-> 
|^| { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } ) )
2221fveq1d 5867 . . 3  |-  ( W  e. AssAlg  ->  ( A `  S )  =  ( ( s  e.  ~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) `  S ) )
2322adantr 465 . 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 461 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  V )
2517elpw2 4611 . . . 4  |-  ( S  e.  ~P V  <->  S  C_  V
)
2624, 25sylibr 212 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  e.  ~P V )
27 assarng 17756 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  Ring )
283subrgid 17226 . . . . . . 7  |-  ( W  e.  Ring  ->  V  e.  (SubRing `  W )
)
2927, 28syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  (SubRing `  W ) )
30 assalmod 17755 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
313, 8lss1 17380 . . . . . . 7  |-  ( W  e.  LMod  ->  V  e.  L )
3230, 31syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  L
)
3329, 32elind 3688 . . . . 5  |-  ( W  e. AssAlg  ->  V  e.  ( (SubRing `  W )  i^i  L ) )
34 sseq2 3526 . . . . . 6  |-  ( t  =  V  ->  ( S  C_  t  <->  S  C_  V
) )
3534rspcev 3214 . . . . 5  |-  ( ( V  e.  ( (SubRing `  W )  i^i  L
)  /\  S  C_  V
)  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
3633, 35sylan 471 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
37 intexrab 4606 . . . 4  |-  ( E. t  e.  ( (SubRing `  W )  i^i  L
) S  C_  t  <->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  S  C_  t }  e.  _V )
3836, 37sylib 196 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t }  e.  _V )
39 sseq1 3525 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  t  <->  S  C_  t
) )
4039rabbidv 3105 . . . . 5  |-  ( s  =  S  ->  { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t }  =  { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
4140inteqd 4287 . . . 4  |-  ( s  =  S  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t }  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
42 eqid 2467 . . . 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 } )
4341, 42fvmptg 5947 . . 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 } )
4426, 38, 43syl2anc 661 . 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 } )
4523, 44eqtrd 2508 1  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   |^|cint 4282    |-> cmpt 4505   ` cfv 5587   Basecbs 14489   Ringcrg 16995  SubRingcsubrg 17220   LModclmod 17307   LSubSpclss 17373  AssAlgcasa 17745  AlgSpancasp 17746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-0g 14696  df-mnd 15731  df-grp 15864  df-mgp 16941  df-ur 16953  df-rng 16997  df-subrg 17222  df-lmod 17309  df-lss 17374  df-assa 17748  df-asp 17749
This theorem is referenced by:  asplss  17765  aspid  17766  aspsubrg  17767  aspss  17768  aspssid  17769  aspval2  17783
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