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Theorem issubassa2 16393
Description: A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
Hypotheses
Ref Expression
issubassa2.a  |-  A  =  (algSc `  W )
issubassa2.l  |-  L  =  ( LSubSp `  W )
Assertion
Ref Expression
issubassa2  |-  ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  ->  ( S  e.  L  <->  ran  A  C_  S
) )

Proof of Theorem issubassa2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issubassa2.a . . . . 5  |-  A  =  (algSc `  W )
2 eqid 2435 . . . . 5  |-  ( 1r
`  W )  =  ( 1r `  W
)
3 eqid 2435 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
41, 2, 3rnascl 16391 . . . 4  |-  ( W  e. AssAlg  ->  ran  A  =  ( ( LSpan `  W
) `  { ( 1r `  W ) } ) )
54ad2antrr 707 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  ran  A  =  ( ( LSpan `  W ) `  {
( 1r `  W
) } ) )
6 issubassa2.l . . . 4  |-  L  =  ( LSubSp `  W )
7 assalmod 16369 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  LMod )
87ad2antrr 707 . . . 4  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  W  e.  LMod )
9 simpr 448 . . . 4  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  S  e.  L )
102subrg1cl 15866 . . . . 5  |-  ( S  e.  (SubRing `  W
)  ->  ( 1r `  W )  e.  S
)
1110ad2antlr 708 . . . 4  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  ( 1r `  W )  e.  S )
126, 3, 8, 9, 11lspsnel5a 16062 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  (
( LSpan `  W ) `  { ( 1r `  W ) } ) 
C_  S )
135, 12eqsstrd 3374 . 2  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  ran  A 
C_  S )
14 subrgsubg 15864 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  S  e.  (SubGrp `  W ) )
1514ad2antlr 708 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  S  e.  (SubGrp `  W )
)
16 simplll 735 . . . . . 6  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  ->  W  e. AssAlg )
17 simprl 733 . . . . . 6  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  ->  x  e.  ( Base `  (Scalar `  W )
) )
18 eqid 2435 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
1918subrgss 15859 . . . . . . . . 9  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
2019ad2antlr 708 . . . . . . . 8  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  S  C_  ( Base `  W
) )
2120sselda 3340 . . . . . . 7  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  y  e.  S
)  ->  y  e.  ( Base `  W )
)
2221adantrl 697 . . . . . 6  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  -> 
y  e.  ( Base `  W ) )
23 eqid 2435 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
24 eqid 2435 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
25 eqid 2435 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
26 eqid 2435 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
271, 23, 24, 18, 25, 26asclmul1 16388 . . . . . 6  |-  ( ( W  e. AssAlg  /\  x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( Base `  W ) )  -> 
( ( A `  x ) ( .r
`  W ) y )  =  ( x ( .s `  W
) y ) )
2816, 17, 22, 27syl3anc 1184 . . . . 5  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  -> 
( ( A `  x ) ( .r
`  W ) y )  =  ( x ( .s `  W
) y ) )
29 simpllr 736 . . . . . 6  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  ->  S  e.  (SubRing `  W
) )
30 simplr 732 . . . . . . . 8  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  x  e.  ( Base `  (Scalar `  W
) ) )  ->  ran  A  C_  S )
311, 23, 24asclfn 16385 . . . . . . . . . 10  |-  A  Fn  ( Base `  (Scalar `  W
) )
3231a1i 11 . . . . . . . . 9  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  A  Fn  ( Base `  (Scalar `  W ) ) )
33 fnfvelrn 5859 . . . . . . . . 9  |-  ( ( A  Fn  ( Base `  (Scalar `  W )
)  /\  x  e.  ( Base `  (Scalar `  W
) ) )  -> 
( A `  x
)  e.  ran  A
)
3432, 33sylan 458 . . . . . . . 8  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  x  e.  ( Base `  (Scalar `  W
) ) )  -> 
( A `  x
)  e.  ran  A
)
3530, 34sseldd 3341 . . . . . . 7  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  x  e.  ( Base `  (Scalar `  W
) ) )  -> 
( A `  x
)  e.  S )
3635adantrr 698 . . . . . 6  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  -> 
( A `  x
)  e.  S )
37 simprr 734 . . . . . 6  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  -> 
y  e.  S )
3825subrgmcl 15870 . . . . . 6  |-  ( ( S  e.  (SubRing `  W
)  /\  ( A `  x )  e.  S  /\  y  e.  S
)  ->  ( ( A `  x )
( .r `  W
) y )  e.  S )
3929, 36, 37, 38syl3anc 1184 . . . . 5  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  -> 
( ( A `  x ) ( .r
`  W ) y )  e.  S )
4028, 39eqeltrrd 2510 . . . 4  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  -> 
( x ( .s
`  W ) y )  e.  S )
4140ralrimivva 2790 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  A. x  e.  ( Base `  (Scalar `  W ) ) A. y  e.  S  (
x ( .s `  W ) y )  e.  S )
4223, 24, 18, 26, 6islss4 16028 . . . . 5  |-  ( W  e.  LMod  ->  ( S  e.  L  <->  ( S  e.  (SubGrp `  W )  /\  A. x  e.  (
Base `  (Scalar `  W
) ) A. y  e.  S  ( x
( .s `  W
) y )  e.  S ) ) )
437, 42syl 16 . . . 4  |-  ( W  e. AssAlg  ->  ( S  e.  L  <->  ( S  e.  (SubGrp `  W )  /\  A. x  e.  (
Base `  (Scalar `  W
) ) A. y  e.  S  ( x
( .s `  W
) y )  e.  S ) ) )
4443ad2antrr 707 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  ( S  e.  L  <->  ( S  e.  (SubGrp `  W )  /\  A. x  e.  (
Base `  (Scalar `  W
) ) A. y  e.  S  ( x
( .s `  W
) y )  e.  S ) ) )
4515, 41, 44mpbir2and 889 . 2  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  S  e.  L )
4613, 45impbida 806 1  |-  ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  ->  ( S  e.  L  <->  ran  A  C_  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   {csn 3806   ran crn 4871    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   Basecbs 13459   .rcmulr 13520  Scalarcsca 13522   .scvsca 13523  SubGrpcsubg 14928   1rcur 15652  SubRingcsubrg 15854   LModclmod 15940   LSubSpclss 15998   LSpanclspn 16037  AssAlgcasa 16359  algSccascl 16361
This theorem is referenced by:  aspval2  16395
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-0g 13717  df-mnd 14680  df-grp 14802  df-minusg 14803  df-sbg 14804  df-subg 14931  df-mgp 15639  df-rng 15653  df-ur 15655  df-subrg 15856  df-lmod 15942  df-lss 15999  df-lsp 16038  df-assa 16362  df-ascl 16364
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