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Theorem aspval2 17975
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 3672 . . . . . . . . 9  |-  ( x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  <->  ( x  e.  (SubRing `  W )  /\  x  e.  ( LSubSp `
 W ) ) )
21anbi1i 695 . . . . . . . 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 649 . . . . . . . 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 249 . . . . . . 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 2443 . . . . . . . . . . 11  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
75, 6issubassa2 17973 . . . . . . . . . 10  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( x  e.  ( LSubSp `  W )  <->  ran 
C  C_  x )
)
87anbi1d 704 . . . . . . . . 9  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( (
x  e.  ( LSubSp `  W )  /\  S  C_  x )  <->  ( ran  C 
C_  x  /\  S  C_  x ) ) )
9 unss 3663 . . . . . . . . 9  |-  ( ( ran  C  C_  x  /\  S  C_  x )  <-> 
( ran  C  u.  S )  C_  x
)
108, 9syl6bb 261 . . . . . . . 8  |-  ( ( W  e. AssAlg  /\  x  e.  (SubRing `  W )
)  ->  ( (
x  e.  ( LSubSp `  W )  /\  S  C_  x )  <->  ( ran  C  u.  S )  C_  x ) )
1110pm5.32da 641 . . . . . . 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 257 . . . . . 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 2579 . . . . 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 465 . . . 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 2802 . . . 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 2802 . . . 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 2509 . . 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 4276 . 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 17956 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }
)
22 assaring 17948 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  Ring )
2320subrgmre 17432 . . . . 5  |-  ( W  e.  Ring  ->  (SubRing `  W
)  e.  (Moore `  V ) )
2422, 23syl 16 . . . 4  |-  ( W  e. AssAlg  ->  (SubRing `  W )  e.  (Moore `  V )
)
2524adantr 465 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  (SubRing `  W )  e.  (Moore `  V ) )
26 eqid 2443 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
27 assalmod 17947 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
28 eqid 2443 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
295, 26, 22, 27, 28, 20asclf 17965 . . . . . 6  |-  ( W  e. AssAlg  ->  C : (
Base `  (Scalar `  W
) ) --> V )
30 frn 5727 . . . . . 6  |-  ( C : ( Base `  (Scalar `  W ) ) --> V  ->  ran  C  C_  V
)
3129, 30syl 16 . . . . 5  |-  ( W  e. AssAlg  ->  ran  C  C_  V
)
3231adantr 465 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ran  C 
C_  V )
33 simpr 461 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  S  C_  V )
3432, 33unssd 3665 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( ran  C  u.  S ) 
C_  V )
35 aspval2.r . . . 4  |-  R  =  (mrCls `  (SubRing `  W
) )
3635mrcval 14989 . . 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 661 . 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 2494 1  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  ( R `  ( ran  C  u.  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   {cab 2428   {crab 2797    u. cun 3459    i^i cin 3460    C_ wss 3461   |^|cint 4271   ran crn 4990   -->wf 5574   ` cfv 5578   Basecbs 14614  Scalarcsca 14682  Moorecmre 14961  mrClscmrc 14962   Ringcrg 17177  SubRingcsubrg 17404   LSubSpclss 17557  AssAlgcasa 17937  AlgSpancasp 17938  algSccascl 17939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-0g 14821  df-mre 14965  df-mrc 14966  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-sbg 16038  df-subg 16177  df-mgp 17121  df-ur 17133  df-ring 17179  df-subrg 17406  df-lmod 17493  df-lss 17558  df-lsp 17597  df-assa 17940  df-asp 17941  df-ascl 17942
This theorem is referenced by:  evlseu  18164
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