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Theorem aspsubrg 16390
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 2436 . . 3  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
41, 2, 3aspval 16387 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }
)
5 ssrab2 3428 . . . 4  |-  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  C_  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )
6 inss1 3561 . . . 4  |-  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) ) 
C_  (SubRing `  W )
75, 6sstri 3357 . . 3  |-  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  C_  (SubRing `  W )
8 fvex 5742 . . . . 5  |-  ( A `
 S )  e. 
_V
94, 8syl6eqelr 2525 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  e.  _V )
10 intex 4356 . . . 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 204 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  =/=  (/) )
12 subrgint 15890 . . 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 645 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  e.  (SubRing `  W )
)
144, 13eqeltrd 2510 1  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  e.  (SubRing `  W )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   |^|cint 4050   ` cfv 5454   Basecbs 13469  SubRingcsubrg 15864   LSubSpclss 16008  AssAlgcasa 16369  AlgSpancasp 16370
This theorem is referenced by:  mplbas2  16531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-subrg 15866  df-lmod 15952  df-lss 16009  df-assa 16372  df-asp 16373
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