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Theorem issubassa 17841
Description: The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
issubassa.s  |-  S  =  ( Ws  A )
issubassa.l  |-  L  =  ( LSubSp `  W )
issubassa.v  |-  V  =  ( Base `  W
)
issubassa.o  |-  .1.  =  ( 1r `  W )
Assertion
Ref Expression
issubassa  |-  ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V
)  ->  ( S  e. AssAlg  <-> 
( A  e.  (SubRing `  W )  /\  A  e.  L ) ) )

Proof of Theorem issubassa
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 999 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  W  e. AssAlg )
2 assaring 17837 . . . . . 6  |-  ( W  e. AssAlg  ->  W  e.  Ring )
31, 2syl 16 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  W  e.  Ring )
4 issubassa.s . . . . . 6  |-  S  =  ( Ws  A )
5 assaring 17837 . . . . . . 7  |-  ( S  e. AssAlg  ->  S  e.  Ring )
65adantl 466 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  Ring )
74, 6syl5eqelr 2560 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( Ws  A
)  e.  Ring )
83, 7jca 532 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( W  e.  Ring  /\  ( Ws  A
)  e.  Ring )
)
9 simpl3 1001 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  C_  V
)
10 simpl2 1000 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  .1.  e.  A )
119, 10jca 532 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( A  C_  V  /\  .1.  e.  A ) )
12 issubassa.v . . . . 5  |-  V  =  ( Base `  W
)
13 issubassa.o . . . . 5  |-  .1.  =  ( 1r `  W )
1412, 13issubrg 17298 . . . 4  |-  ( A  e.  (SubRing `  W
)  <->  ( ( W  e.  Ring  /\  ( Ws  A )  e.  Ring )  /\  ( A  C_  V  /\  .1.  e.  A
) ) )
158, 11, 14sylanbrc 664 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  (SubRing `  W ) )
16 assalmod 17836 . . . . 5  |-  ( S  e. AssAlg  ->  S  e.  LMod )
1716adantl 466 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  LMod )
18 assalmod 17836 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  LMod )
19 issubassa.l . . . . . 6  |-  L  =  ( LSubSp `  W )
204, 12, 19islss3 17474 . . . . 5  |-  ( W  e.  LMod  ->  ( A  e.  L  <->  ( A  C_  V  /\  S  e. 
LMod ) ) )
211, 18, 203syl 20 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( A  e.  L  <->  ( A  C_  V  /\  S  e.  LMod ) ) )
229, 17, 21mpbir2and 920 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  L )
2315, 22jca 532 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( A  e.  (SubRing `  W )  /\  A  e.  L
) )
2412subrgss 17299 . . . . . 6  |-  ( A  e.  (SubRing `  W
)  ->  A  C_  V
)
2524ad2antrl 727 . . . . 5  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  A  C_  V )
264, 12ressbas2 14562 . . . . 5  |-  ( A 
C_  V  ->  A  =  ( Base `  S
) )
2725, 26syl 16 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  A  =  ( Base `  S ) )
28 eqid 2467 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
294, 28resssca 14649 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  (Scalar `  W
)  =  (Scalar `  S ) )
3029ad2antrl 727 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  =  (Scalar `  S )
)
31 eqidd 2468 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
32 eqid 2467 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
334, 32ressvsca 14650 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .s `  W )  =  ( .s `  S ) )
3433ad2antrl 727 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .s `  W
)  =  ( .s
`  S ) )
35 eqid 2467 . . . . . 6  |-  ( .r
`  W )  =  ( .r `  W
)
364, 35ressmulr 14624 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  S ) )
3736ad2antrl 727 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .r `  W
)  =  ( .r
`  S ) )
38 simpr 461 . . . . 5  |-  ( ( A  e.  (SubRing `  W
)  /\  A  e.  L )  ->  A  e.  L )
394, 19lsslmod 17475 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  L )  ->  S  e.  LMod )
4018, 38, 39syl2an 477 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  LMod )
414subrgring 17301 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  S  e.  Ring )
4241ad2antrl 727 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  Ring )
4328assasca 17838 . . . . 5  |-  ( W  e. AssAlg  ->  (Scalar `  W )  e.  CRing )
4443adantr 465 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  e.  CRing )
45 simpll 753 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  ->  W  e. AssAlg )
46 simpr1 1002 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  ->  x  e.  ( Base `  (Scalar `  W )
) )
4725adantr 465 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  ->  A  C_  V )
48 simpr2 1003 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  -> 
y  e.  A )
4947, 48sseldd 3510 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  -> 
y  e.  V )
50 simpr3 1004 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  -> 
z  e.  A )
5147, 50sseldd 3510 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  -> 
z  e.  V )
52 eqid 2467 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
5312, 28, 52, 32, 35assaass 17834 . . . . 5  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  y  e.  V  /\  z  e.  V
) )  ->  (
( x ( .s
`  W ) y ) ( .r `  W ) z )  =  ( x ( .s `  W ) ( y ( .r
`  W ) z ) ) )
5445, 46, 49, 51, 53syl13anc 1230 . . . 4  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x ( .s `  W ) y ) ( .r
`  W ) z )  =  ( x ( .s `  W
) ( y ( .r `  W ) z ) ) )
5512, 28, 52, 32, 35assaassr 17835 . . . . 5  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  y  e.  V  /\  z  e.  V
) )  ->  (
y ( .r `  W ) ( x ( .s `  W
) z ) )  =  ( x ( .s `  W ) ( y ( .r
`  W ) z ) ) )
5645, 46, 49, 51, 55syl13anc 1230 . . . 4  |-  ( ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  A  /\  z  e.  A ) )  -> 
( y ( .r
`  W ) ( x ( .s `  W ) z ) )  =  ( x ( .s `  W
) ( y ( .r `  W ) z ) ) )
5727, 30, 31, 34, 37, 40, 42, 44, 54, 56isassad 17840 . . 3  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
58573ad2antl1 1158 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
5923, 58impbida 830 1  |-  ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V
)  ->  ( S  e. AssAlg  <-> 
( A  e.  (SubRing `  W )  /\  A  e.  L ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   ` cfv 5594  (class class class)co 6295   Basecbs 14506   ↾s cress 14507   .rcmulr 14572  Scalarcsca 14574   .scvsca 14575   1rcur 17023   Ringcrg 17068   CRingccrg 17069  SubRingcsubrg 17294   LModclmod 17381   LSubSpclss 17447  AssAlgcasa 17826
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-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-1st 6795  df-2nd 6796  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-4 10608  df-5 10609  df-6 10610  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-sca 14587  df-vsca 14588  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069  df-mgp 17012  df-ur 17024  df-ring 17070  df-subrg 17296  df-lmod 17383  df-lss 17448  df-assa 17829
This theorem is referenced by:  mplassa  17984  ply1assa  18106
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