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Theorem issubassa2 16331
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 2388 . . . . 5  |-  ( 1r
`  W )  =  ( 1r `  W
)
3 eqid 2388 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
41, 2, 3rnascl 16329 . . . 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 16307 . . . . 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 15804 . . . . 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 16000 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  (
( LSpan `  W ) `  { ( 1r `  W ) } ) 
C_  S )
135, 12eqsstrd 3326 . 2  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  ran  A 
C_  S )
14 subrgsubg 15802 . . . 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 2388 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
1918subrgss 15797 . . . . . . . . 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 3292 . . . . . . 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 2388 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
24 eqid 2388 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
25 eqid 2388 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
26 eqid 2388 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
271, 23, 24, 18, 25, 26asclmul1 16326 . . . . . 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 16323 . . . . . . . . . 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 5807 . . . . . . . . 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 3293 . . . . . . 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 15808 . . . . . 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 2463 . . . 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 2742 . . 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 15966 . . . . 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 1649    e. wcel 1717   A.wral 2650    C_ wss 3264   {csn 3758   ran crn 4820    Fn wfn 5390   ` cfv 5395  (class class class)co 6021   Basecbs 13397   .rcmulr 13458  Scalarcsca 13460   .scvsca 13461  SubGrpcsubg 14866   1rcur 15590  SubRingcsubrg 15792   LModclmod 15878   LSubSpclss 15936   LSpanclspn 15975  AssAlgcasa 16297  algSccascl 16299
This theorem is referenced by:  aspval2  16333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-mgp 15577  df-rng 15591  df-ur 15593  df-subrg 15794  df-lmod 15880  df-lss 15937  df-lsp 15976  df-assa 16300  df-ascl 16302
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