MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  aspval Structured version   Unicode version

Theorem aspval 16387
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 5728 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 aspval.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2486 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  V )
54pweqd 3804 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
6 fveq2 5728 . . . . . . . . . 10  |-  ( w  =  W  ->  (SubRing `  w )  =  (SubRing `  W ) )
7 fveq2 5728 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
8 aspval.l . . . . . . . . . . 11  |-  L  =  ( LSubSp `  W )
97, 8syl6eqr 2486 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  L )
106, 9ineq12d 3543 . . . . . . . . 9  |-  ( w  =  W  ->  (
(SubRing `  w )  i^i  ( LSubSp `  w )
)  =  ( (SubRing `  W )  i^i  L
) )
11 rabeq 2950 . . . . . . . . 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 4055 . . . . . . 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 4287 . . . . . 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 16373 . . . . . 6  |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
16 fvex 5742 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
173, 16eqeltri 2506 . . . . . . . 8  |-  V  e. 
_V
1817pwex 4382 . . . . . . 7  |-  ~P V  e.  _V
1918mptex 5966 . . . . . 6  |-  ( s  e.  ~P V  |->  |^|
{ t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } )  e.  _V
2014, 15, 19fvmpt 5806 . . . . 5  |-  ( W  e. AssAlg  ->  (AlgSpan `  W )  =  ( s  e. 
~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) )
211, 20syl5eq 2480 . . . 4  |-  ( W  e. AssAlg  ->  A  =  ( s  e.  ~P V  |-> 
|^| { t  e.  ( (SubRing `  W )  i^i  L )  |  s 
C_  t } ) )
2221fveq1d 5730 . . 3  |-  ( W  e. AssAlg  ->  ( A `  S )  =  ( ( s  e.  ~P V  |->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t } ) `  S ) )
2322adantr 452 . 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 448 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  V )
2517elpw2 4364 . . . 4  |-  ( S  e.  ~P V  <->  S  C_  V
)
2624, 25sylibr 204 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  e.  ~P V )
27 assarng 16380 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  Ring )
283subrgid 15870 . . . . . . 7  |-  ( W  e.  Ring  ->  V  e.  (SubRing `  W )
)
2927, 28syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  (SubRing `  W ) )
30 assalmod 16379 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
313, 8lss1 16015 . . . . . . 7  |-  ( W  e.  LMod  ->  V  e.  L )
3230, 31syl 16 . . . . . 6  |-  ( W  e. AssAlg  ->  V  e.  L
)
33 elin 3530 . . . . . 6  |-  ( V  e.  ( (SubRing `  W
)  i^i  L )  <->  ( V  e.  (SubRing `  W
)  /\  V  e.  L ) )
3429, 32, 33sylanbrc 646 . . . . 5  |-  ( W  e. AssAlg  ->  V  e.  ( (SubRing `  W )  i^i  L ) )
35 sseq2 3370 . . . . . 6  |-  ( t  =  V  ->  ( S  C_  t  <->  S  C_  V
) )
3635rspcev 3052 . . . . 5  |-  ( ( V  e.  ( (SubRing `  W )  i^i  L
)  /\  S  C_  V
)  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
3734, 36sylan 458 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  E. t  e.  ( (SubRing `  W
)  i^i  L ) S  C_  t )
38 intexrab 4359 . . . 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 189 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t }  e.  _V )
40 sseq1 3369 . . . . . 6  |-  ( s  =  S  ->  (
s  C_  t  <->  S  C_  t
) )
4140rabbidv 2948 . . . . 5  |-  ( s  =  S  ->  { t  e.  ( (SubRing `  W
)  i^i  L )  |  s  C_  t }  =  { t  e.  ( (SubRing `  W
)  i^i  L )  |  S  C_  t } )
4241inteqd 4055 . . . 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 2436 . . . 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 5804 . . 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 643 . 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 2468 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 359    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   |^|cint 4050    e. cmpt 4266   ` cfv 5454   Basecbs 13469   Ringcrg 15660  SubRingcsubrg 15864   LModclmod 15950   LSubSpclss 16008  AssAlgcasa 16369  AlgSpancasp 16370
This theorem is referenced by:  asplss  16388  aspid  16389  aspsubrg  16390  aspss  16391  aspssid  16392  aspval2  16405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-mgp 15649  df-rng 15663  df-ur 15665  df-subrg 15866  df-lmod 15952  df-lss 16009  df-assa 16372  df-asp 16373
  Copyright terms: Public domain W3C validator