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Theorem aspsubrg 17848
Description: The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
aspval.a  |-  A  =  (AlgSpan `  W )
aspval.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
aspsubrg  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  e.  (SubRing `  W )
)

Proof of Theorem aspsubrg
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 aspval.a . . 3  |-  A  =  (AlgSpan `  W )
2 aspval.v . . 3  |-  V  =  ( Base `  W
)
3 eqid 2467 . . 3  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
41, 2, 3aspval 17845 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }
)
5 ssrab2 3590 . . . 4  |-  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  C_  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )
6 inss1 3723 . . . 4  |-  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) ) 
C_  (SubRing `  W )
75, 6sstri 3518 . . 3  |-  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  C_  (SubRing `  W )
8 fvex 5882 . . . . 5  |-  ( A `
 S )  e. 
_V
94, 8syl6eqelr 2564 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  e.  _V )
10 intex 4609 . . . 4  |-  ( { t  e.  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) )  |  S  C_  t }  =/=  (/)  <->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  e.  _V )
119, 10sylibr 212 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  =/=  (/) )
12 subrgint 17320 . . 3  |-  ( ( { t  e.  ( (SubRing `  W )  i^i  ( LSubSp `  W )
)  |  S  C_  t }  C_  (SubRing `  W
)  /\  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  =/=  (/) )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  e.  (SubRing `  W )
)
137, 11, 12sylancr 663 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  e.  (SubRing `  W )
)
144, 13eqeltrd 2555 1  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  e.  (SubRing `  W )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   |^|cint 4288   ` cfv 5594   Basecbs 14506  SubRingcsubrg 17294   LSubSpclss 17447  AssAlgcasa 17826  AlgSpancasp 17827
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-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 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-subg 16069  df-mgp 17012  df-ur 17024  df-ring 17070  df-subrg 17296  df-lmod 17383  df-lss 17448  df-assa 17829  df-asp 17830
This theorem is referenced by:  mplbas2  18002  mplbas2OLD  18003
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