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Theorem aspval2 16325
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 3466 . . . . . . . . 9  |-  ( x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  <->  ( x  e.  (SubRing `  W )  /\  x  e.  ( LSubSp `
 W ) ) )
21anbi1i 677 . . . . . . . 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 631 . . . . . . . 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 241 . . . . . . 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 2380 . . . . . . . . . . 11  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
75, 6issubassa2 16323 . . . . . . . . . 10  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( x  e.  ( LSubSp `  W )  <->  ran 
C  C_  x )
)
87anbi1d 686 . . . . . . . . 9  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( (
x  e.  ( LSubSp `  W )  /\  S  C_  x )  <->  ( ran  C 
C_  x  /\  S  C_  x ) ) )
9 unss 3457 . . . . . . . . 9  |-  ( ( ran  C  C_  x  /\  S  C_  x )  <-> 
( ran  C  u.  S )  C_  x
)
108, 9syl6bb 253 . . . . . . . 8  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( (
x  e.  ( LSubSp `  W )  /\  S  C_  x )  <->  ( ran  C  u.  S )  C_  x ) )
1110pm5.32da 623 . . . . . . 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 249 . . . . . 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 2494 . . . . 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 452 . . . 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 2651 . . . 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 2651 . . . 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 2437 . . 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 3990 . 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 16307 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }
)
22 assarng 16300 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  Ring )
2320subrgmre 15812 . . . . 5  |-  ( W  e.  Ring  ->  (SubRing `  W
)  e.  (Moore `  V ) )
2422, 23syl 16 . . . 4  |-  ( W  e. AssAlg  ->  (SubRing `  W )  e.  (Moore `  V )
)
2524adantr 452 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  (SubRing `  W )  e.  (Moore `  V ) )
26 eqid 2380 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
27 assalmod 16299 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
28 eqid 2380 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
295, 26, 22, 27, 28, 20asclf 16316 . . . . . 6  |-  ( W  e. AssAlg  ->  C : (
Base `  (Scalar `  W
) ) --> V )
30 frn 5530 . . . . . 6  |-  ( C : ( Base `  (Scalar `  W ) ) --> V  ->  ran  C  C_  V
)
3129, 30syl 16 . . . . 5  |-  ( W  e. AssAlg  ->  ran  C  C_  V
)
3231adantr 452 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ran  C 
C_  V )
33 simpr 448 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  V )
3432, 33unssd 3459 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( ran  C  u.  S ) 
C_  V )
35 aspval2.r . . . 4  |-  R  =  (mrCls `  (SubRing `  W
) )
3635mrcval 13755 . . 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 643 . 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 2422 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 359    = wceq 1649    e. wcel 1717   {cab 2366   {crab 2646    u. cun 3254    i^i cin 3255    C_ wss 3256   |^|cint 3985   ran crn 4812   -->wf 5383   ` cfv 5387   Basecbs 13389  Scalarcsca 13452  Moorecmre 13727  mrClscmrc 13728   Ringcrg 15580  SubRingcsubrg 15784   LSubSpclss 15928  AssAlgcasa 16289  AlgSpancasp 16290  algSccascl 16291
This theorem is referenced by:  evlseu  19797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-0g 13647  df-mre 13731  df-mrc 13732  df-mnd 14610  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-mgp 15569  df-rng 15583  df-ur 15585  df-subrg 15786  df-lmod 15872  df-lss 15929  df-lsp 15968  df-assa 16292  df-asp 16293  df-ascl 16294
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