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Theorem issubassa2 17427
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 2443 . . . . 5  |-  ( 1r
`  W )  =  ( 1r `  W
)
3 eqid 2443 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
41, 2, 3rnascl 17425 . . . 4  |-  ( W  e. AssAlg  ->  ran  A  =  ( ( LSpan `  W
) `  { ( 1r `  W ) } ) )
54ad2antrr 725 . . 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 17403 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  LMod )
87ad2antrr 725 . . . 4  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  W  e.  LMod )
9 simpr 461 . . . 4  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  S  e.  L )
102subrg1cl 16885 . . . . 5  |-  ( S  e.  (SubRing `  W
)  ->  ( 1r `  W )  e.  S
)
1110ad2antlr 726 . . . 4  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  ( 1r `  W )  e.  S )
126, 3, 8, 9, 11lspsnel5a 17089 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  (
( LSpan `  W ) `  { ( 1r `  W ) } ) 
C_  S )
135, 12eqsstrd 3402 . 2  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  S  e.  L )  ->  ran  A 
C_  S )
14 subrgsubg 16883 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  S  e.  (SubGrp `  W ) )
1514ad2antlr 726 . . 3  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  S  e.  (SubGrp `  W )
)
16 simplll 757 . . . . . 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 755 . . . . . 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 2443 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
1918subrgss 16878 . . . . . . . . 9  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
2019ad2antlr 726 . . . . . . . 8  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  S  C_  ( Base `  W
) )
2120sselda 3368 . . . . . . 7  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  y  e.  S
)  ->  y  e.  ( Base `  W )
)
2221adantrl 715 . . . . . 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 2443 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
24 eqid 2443 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
25 eqid 2443 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
26 eqid 2443 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
271, 23, 24, 18, 25, 26asclmul1 17422 . . . . . 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 1218 . . . . 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 758 . . . . . 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 754 . . . . . . . 8  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  x  e.  ( Base `  (Scalar `  W
) ) )  ->  ran  A  C_  S )
311, 23, 24asclfn 17419 . . . . . . . . . 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 5852 . . . . . . . . 9  |-  ( ( A  Fn  ( Base `  (Scalar `  W )
)  /\  x  e.  ( Base `  (Scalar `  W
) ) )  -> 
( A `  x
)  e.  ran  A
)
3432, 33sylan 471 . . . . . . . 8  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  x  e.  ( Base `  (Scalar `  W
) ) )  -> 
( A `  x
)  e.  ran  A
)
3530, 34sseldd 3369 . . . . . . 7  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  x  e.  ( Base `  (Scalar `  W
) ) )  -> 
( A `  x
)  e.  S )
3635adantrr 716 . . . . . 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 756 . . . . . 6  |-  ( ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W ) )  /\  ran  A  C_  S )  /\  ( x  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  S ) )  -> 
y  e.  S )
3825subrgmcl 16889 . . . . . 6  |-  ( ( S  e.  (SubRing `  W
)  /\  ( A `  x )  e.  S  /\  y  e.  S
)  ->  ( ( A `  x )
( .r `  W
) y )  e.  S )
3929, 36, 37, 38syl3anc 1218 . . . . 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 2518 . . . 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 2820 . . 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 17055 . . . . 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 725 . . 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 913 . 2  |-  ( ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  /\  ran  A  C_  S )  ->  S  e.  L )
4613, 45impbida 828 1  |-  ( ( W  e. AssAlg  /\  S  e.  (SubRing `  W )
)  ->  ( S  e.  L  <->  ran  A  C_  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727    C_ wss 3340   {csn 3889   ran crn 4853    Fn wfn 5425   ` cfv 5430  (class class class)co 6103   Basecbs 14186   .rcmulr 14251  Scalarcsca 14253   .scvsca 14254  SubGrpcsubg 15687   1rcur 16615  SubRingcsubrg 16873   LModclmod 16960   LSubSpclss 17025   LSpanclspn 17064  AssAlgcasa 17393  algSccascl 17395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-mgp 16604  df-ur 16616  df-rng 16659  df-subrg 16875  df-lmod 16962  df-lss 17026  df-lsp 17065  df-assa 17396  df-ascl 17398
This theorem is referenced by:  aspval2  17429
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