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Theorem aspval 18090
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 5774 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 aspval.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2441 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  V )
54pweqd 3932 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
6 fveq2 5774 . . . . . . . . . 10  |-  ( w  =  W  ->  (SubRing `  w )  =  (SubRing `  W ) )
7 fveq2 5774 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 aspval.l . . . . . . . . . . 11  |-  L  =  ( LSubSp `  W )
97, 8syl6eqr 2441 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
106, 9ineq12d 3615 . . . . . . . . 9  |-  ( w  =  W  ->  (
(SubRing `  w )  i^i  ( LSubSp `  w )
)  =  ( (SubRing `  W )  i^i  L
) )
11 rabeq 3028 . . . . . . . . 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 4204 . . . . . . 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 4445 . . . . . 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 18075 . . . . . 6  |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
16 fvex 5784 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
173, 16eqeltri 2466 . . . . . . . 8  |-  V  e. 
_V
1817pwex 4548 . . . . . . 7  |-  ~P V  e.  _V
1918mptex 6044 . . . . . 6  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )  e.  _V
2014, 15, 19fvmpt 5857 . . . . 5  |-  ( W  e. AssAlg  ->  (AlgSpan `  W )  =  ( s  e. 
~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) )
211, 20syl5eq 2435 . . . 4  |-  ( W  e. AssAlg  ->  A  =  ( s  e.  ~P V  |-> 
|^| { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } ) )
2221fveq1d 5776 . . 3  |-  ( W  e. AssAlg  ->  ( A `  S )  =  ( ( s  e.  ~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) `  S ) )
2322adantr 463 . 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 459 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  V )
2517elpw2 4529 . . . 4  |-  ( S  e.  ~P V  <->  S  C_  V
)
2624, 25sylibr 212 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  e.  ~P V )
27 assaring 18082 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  Ring )
283subrgid 17544 . . . . . . 7  |-  ( W  e.  Ring  ->  V  e.  (SubRing `  W )
)
2927, 28syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  (SubRing `  W ) )
30 assalmod 18081 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
313, 8lss1 17698 . . . . . . 7  |-  ( W  e.  LMod  ->  V  e.  L )
3230, 31syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  L
)
3329, 32elind 3602 . . . . 5  |-  ( W  e. AssAlg  ->  V  e.  ( (SubRing `  W )  i^i  L ) )
34 sseq2 3439 . . . . . 6  |-  ( t  =  V  ->  ( S  C_  t  <->  S  C_  V
) )
3534rspcev 3135 . . . . 5  |-  ( ( V  e.  ( (SubRing `  W )  i^i  L
)  /\  S  C_  V
)  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
3633, 35sylan 469 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
37 intexrab 4524 . . . 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 3438 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  t  <->  S  C_  t
) )
4039rabbidv 3026 . . . . 5  |-  ( s  =  S  ->  { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t }  =  { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
4140inteqd 4204 . . . 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 2382 . . . 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 5855 . . 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 659 . 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 2423 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 367    = wceq 1399    e. wcel 1826   E.wrex 2733   {crab 2736   _Vcvv 3034    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   |^|cint 4199    |-> cmpt 4425   ` cfv 5496   Basecbs 14634   Ringcrg 17311  SubRingcsubrg 17538   LModclmod 17625   LSubSpclss 17691  AssAlgcasa 18071  AlgSpancasp 18072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-mgp 17255  df-ur 17267  df-ring 17313  df-subrg 17540  df-lmod 17627  df-lss 17692  df-assa 18074  df-asp 18075
This theorem is referenced by:  asplss  18091  aspid  18092  aspsubrg  18093  aspss  18094  aspssid  18095  aspval2  18109
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