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Theorem aspval2 16102
Description: The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
aspval2.a  |-  A  =  (AlgSpan `  W )
aspval2.c  |-  C  =  (algSc `  W )
aspval2.r  |-  R  =  (mrCls `  (SubRing `  W
) )
aspval2.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
aspval2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  ( R `  ( ran  C  u.  S
) ) )

Proof of Theorem aspval2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3371 . . . . . . . . 9  |-  ( x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  <->  ( x  e.  (SubRing `  W )  /\  x  e.  ( LSubSp `
 W ) ) )
21anbi1i 676 . . . . . . . 8  |-  ( ( x  e.  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) )  /\  S  C_  x
)  <->  ( ( x  e.  (SubRing `  W
)  /\  x  e.  ( LSubSp `  W )
)  /\  S  C_  x
) )
3 anass 630 . . . . . . . 8  |-  ( ( ( x  e.  (SubRing `  W )  /\  x  e.  ( LSubSp `  W )
)  /\  S  C_  x
)  <->  ( x  e.  (SubRing `  W )  /\  ( x  e.  (
LSubSp `  W )  /\  S  C_  x ) ) )
42, 3bitri 240 . . . . . . 7  |-  ( ( x  e.  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) )  /\  S  C_  x
)  <->  ( x  e.  (SubRing `  W )  /\  ( x  e.  (
LSubSp `  W )  /\  S  C_  x ) ) )
5 aspval2.c . . . . . . . . . . 11  |-  C  =  (algSc `  W )
6 eqid 2296 . . . . . . . . . . 11  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
75, 6issubassa2 16100 . . . . . . . . . 10  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( x  e.  ( LSubSp `  W )  <->  ran 
C  C_  x )
)
87anbi1d 685 . . . . . . . . 9  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( (
x  e.  ( LSubSp `  W )  /\  S  C_  x )  <->  ( ran  C 
C_  x  /\  S  C_  x ) ) )
9 unss 3362 . . . . . . . . 9  |-  ( ( ran  C  C_  x  /\  S  C_  x )  <-> 
( ran  C  u.  S )  C_  x
)
108, 9syl6bb 252 . . . . . . . 8  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( (
x  e.  ( LSubSp `  W )  /\  S  C_  x )  <->  ( ran  C  u.  S )  C_  x ) )
1110pm5.32da 622 . . . . . . 7  |-  ( W  e. AssAlg  ->  ( ( x  e.  (SubRing `  W
)  /\  ( x  e.  ( LSubSp `  W )  /\  S  C_  x ) )  <->  ( x  e.  (SubRing `  W )  /\  ( ran  C  u.  S )  C_  x
) ) )
124, 11syl5bb 248 . . . . . 6  |-  ( W  e. AssAlg  ->  ( ( x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  /\  S  C_  x )  <->  ( x  e.  (SubRing `  W )  /\  ( ran  C  u.  S )  C_  x
) ) )
1312abbidv 2410 . . . . 5  |-  ( W  e. AssAlg  ->  { x  |  ( x  e.  ( (SubRing `  W )  i^i  ( LSubSp `  W )
)  /\  S  C_  x
) }  =  {
x  |  ( x  e.  (SubRing `  W
)  /\  ( ran  C  u.  S )  C_  x ) } )
1413adantr 451 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  { x  |  ( x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  /\  S  C_  x ) }  =  { x  |  ( x  e.  (SubRing `  W )  /\  ( ran  C  u.  S ) 
C_  x ) } )
15 df-rab 2565 . . . 4  |-  { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }  =  { x  |  ( x  e.  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) )  /\  S  C_  x
) }
16 df-rab 2565 . . . 4  |-  { x  e.  (SubRing `  W )  |  ( ran  C  u.  S )  C_  x }  =  { x  |  ( x  e.  (SubRing `  W )  /\  ( ran  C  u.  S )  C_  x
) }
1714, 15, 163eqtr4g 2353 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }  =  { x  e.  (SubRing `  W )  |  ( ran  C  u.  S
)  C_  x }
)
1817inteqd 3883 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }  =  |^| { x  e.  (SubRing `  W )  |  ( ran  C  u.  S )  C_  x } )
19 aspval2.a . . 3  |-  A  =  (AlgSpan `  W )
20 aspval2.v . . 3  |-  V  =  ( Base `  W
)
2119, 20, 6aspval 16084 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }
)
22 assarng 16077 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  Ring )
2320subrgmre 15585 . . . . 5  |-  ( W  e.  Ring  ->  (SubRing `  W
)  e.  (Moore `  V ) )
2422, 23syl 15 . . . 4  |-  ( W  e. AssAlg  ->  (SubRing `  W )  e.  (Moore `  V )
)
2524adantr 451 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  (SubRing `  W )  e.  (Moore `  V ) )
26 eqid 2296 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
27 assalmod 16076 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
28 eqid 2296 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
295, 26, 22, 27, 28, 20asclf 16093 . . . . . 6  |-  ( W  e. AssAlg  ->  C : (
Base `  (Scalar `  W
) ) --> V )
30 frn 5411 . . . . . 6  |-  ( C : ( Base `  (Scalar `  W ) ) --> V  ->  ran  C  C_  V
)
3129, 30syl 15 . . . . 5  |-  ( W  e. AssAlg  ->  ran  C  C_  V
)
3231adantr 451 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ran  C 
C_  V )
33 simpr 447 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  V )
3432, 33unssd 3364 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( ran  C  u.  S ) 
C_  V )
35 aspval2.r . . . 4  |-  R  =  (mrCls `  (SubRing `  W
) )
3635mrcval 13528 . . 3  |-  ( ( (SubRing `  W )  e.  (Moore `  V )  /\  ( ran  C  u.  S )  C_  V
)  ->  ( R `  ( ran  C  u.  S ) )  = 
|^| { x  e.  (SubRing `  W )  |  ( ran  C  u.  S
)  C_  x }
)
3725, 34, 36syl2anc 642 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( R `  ( ran  C  u.  S ) )  =  |^| { x  e.  (SubRing `  W )  |  ( ran  C  u.  S )  C_  x } )
3818, 21, 373eqtr4d 2338 1  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  ( R `  ( ran  C  u.  S
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    u. cun 3163    i^i cin 3164    C_ wss 3165   |^|cint 3878   ran crn 4706   -->wf 5267   ` cfv 5271   Basecbs 13164  Scalarcsca 13227  Moorecmre 13500  mrClscmrc 13501   Ringcrg 15353  SubRingcsubrg 15557   LSubSpclss 15705  AssAlgcasa 16066  AlgSpancasp 16067  algSccascl 16068
This theorem is referenced by:  evlseu  19416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mre 13504  df-mrc 13505  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-assa 16069  df-asp 16070  df-ascl 16071
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