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Theorem aspval 17845
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 5852 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 aspval.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2500 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  V )
54pweqd 3998 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
6 fveq2 5852 . . . . . . . . . 10  |-  ( w  =  W  ->  (SubRing `  w )  =  (SubRing `  W ) )
7 fveq2 5852 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 aspval.l . . . . . . . . . . 11  |-  L  =  ( LSubSp `  W )
97, 8syl6eqr 2500 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
106, 9ineq12d 3683 . . . . . . . . 9  |-  ( w  =  W  ->  (
(SubRing `  w )  i^i  ( LSubSp `  w )
)  =  ( (SubRing `  W )  i^i  L
) )
11 rabeq 3087 . . . . . . . . 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 4272 . . . . . . 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 4511 . . . . . 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 17830 . . . . . 6  |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
16 fvex 5862 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
173, 16eqeltri 2525 . . . . . . . 8  |-  V  e. 
_V
1817pwex 4616 . . . . . . 7  |-  ~P V  e.  _V
1918mptex 6124 . . . . . 6  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )  e.  _V
2014, 15, 19fvmpt 5937 . . . . 5  |-  ( W  e. AssAlg  ->  (AlgSpan `  W )  =  ( s  e. 
~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) )
211, 20syl5eq 2494 . . . 4  |-  ( W  e. AssAlg  ->  A  =  ( s  e.  ~P V  |-> 
|^| { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } ) )
2221fveq1d 5854 . . 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 4597 . . . 4  |-  ( S  e.  ~P V  <->  S  C_  V
)
2624, 25sylibr 212 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  e.  ~P V )
27 assaring 17837 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  Ring )
283subrgid 17299 . . . . . . 7  |-  ( W  e.  Ring  ->  V  e.  (SubRing `  W )
)
2927, 28syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  (SubRing `  W ) )
30 assalmod 17836 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
313, 8lss1 17453 . . . . . . 7  |-  ( W  e.  LMod  ->  V  e.  L )
3230, 31syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  L
)
3329, 32elind 3670 . . . . 5  |-  ( W  e. AssAlg  ->  V  e.  ( (SubRing `  W )  i^i  L ) )
34 sseq2 3508 . . . . . 6  |-  ( t  =  V  ->  ( S  C_  t  <->  S  C_  V
) )
3534rspcev 3194 . . . . 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 4592 . . . 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 3507 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  t  <->  S  C_  t
) )
4039rabbidv 3085 . . . . 5  |-  ( s  =  S  ->  { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t }  =  { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
4140inteqd 4272 . . . 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 2441 . . . 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 5935 . . 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 2482 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 1381    e. wcel 1802   E.wrex 2792   {crab 2795   _Vcvv 3093    i^i cin 3457    C_ wss 3458   ~Pcpw 3993   |^|cint 4267    |-> cmpt 4491   ` cfv 5574   Basecbs 14504   Ringcrg 17066  SubRingcsubrg 17293   LModclmod 17380   LSubSpclss 17446  AssAlgcasa 17826  AlgSpancasp 17827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-mgp 17010  df-ur 17022  df-ring 17068  df-subrg 17295  df-lmod 17382  df-lss 17447  df-assa 17829  df-asp 17830
This theorem is referenced by:  asplss  17846  aspid  17847  aspsubrg  17848  aspss  17849  aspssid  17850  aspval2  17864
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