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Theorem aspval2 17866
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 3692 . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
75, 6issubassa2 17864 . . . . . . . . . 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 3683 . . . . . . . . 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 2603 . . . . 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 2826 . . . 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 2826 . . . 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 2533 . . 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 4293 . 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 17847 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }
)
22 assaring 17839 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  Ring )
2320subrgmre 17324 . . . . 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 2467 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
27 assalmod 17838 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
28 eqid 2467 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
295, 26, 22, 27, 28, 20asclf 17856 . . . . . 6  |-  ( W  e. AssAlg  ->  C : (
Base `  (Scalar `  W
) ) --> V )
30 frn 5743 . . . . . 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 3685 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( ran  C  u.  S ) 
C_  V )
35 aspval2.r . . . 4  |-  R  =  (mrCls `  (SubRing `  W
) )
3635mrcval 14882 . . 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 2518 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 1379    e. wcel 1767   {cab 2452   {crab 2821    u. cun 3479    i^i cin 3480    C_ wss 3481   |^|cint 4288   ran crn 5006   -->wf 5590   ` cfv 5594   Basecbs 14507  Scalarcsca 14575  Moorecmre 14854  mrClscmrc 14855   Ringcrg 17070  SubRingcsubrg 17296   LSubSpclss 17449  AssAlgcasa 17828  AlgSpancasp 17829  algSccascl 17830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-0g 14714  df-mre 14858  df-mrc 14859  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-mgp 17014  df-ur 17026  df-ring 17072  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-assa 17831  df-asp 17832  df-ascl 17833
This theorem is referenced by:  evlseu  18055
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