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Theorem aspval2 16086
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 3358 . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
75, 6issubassa2 16084 . . . . . . . . . 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 3349 . . . . . . . . 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 2397 . . . . 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 2552 . . . 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 2552 . . . 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 2340 . . 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 3867 . 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 16068 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { x  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  x }
)
22 assarng 16061 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  Ring )
2320subrgmre 15569 . . . . 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 2283 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
27 assalmod 16060 . . . . . . 7  |-  ( W  e. AssAlg  ->  W  e.  LMod )
28 eqid 2283 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
295, 26, 22, 27, 28, 20asclf 16077 . . . . . 6  |-  ( W  e. AssAlg  ->  C : (
Base `  (Scalar `  W
) ) --> V )
30 frn 5395 . . . . . 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 3351 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( ran  C  u.  S ) 
C_  V )
35 aspval2.r . . . 4  |-  R  =  (mrCls `  (SubRing `  W
) )
3635mrcval 13512 . . 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 2325 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 1623    e. wcel 1684   {cab 2269   {crab 2547    u. cun 3150    i^i cin 3151    C_ wss 3152   |^|cint 3862   ran crn 4690   -->wf 5251   ` cfv 5255   Basecbs 13148  Scalarcsca 13211  Moorecmre 13484  mrClscmrc 13485   Ringcrg 15337  SubRingcsubrg 15541   LSubSpclss 15689  AssAlgcasa 16050  AlgSpancasp 16051  algSccascl 16052
This theorem is referenced by:  evlseu  19400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mre 13488  df-mrc 13489  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-assa 16053  df-asp 16054  df-ascl 16055
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