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Theorem issubassa 18291
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 1000 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  W  e. AssAlg )
2 assaring 18287 . . . . . 6  |-  ( W  e. AssAlg  ->  W  e.  Ring )
31, 2syl 17 . . . . 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 18287 . . . . . . 7  |-  ( S  e. AssAlg  ->  S  e.  Ring )
65adantl 464 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  Ring )
74, 6syl5eqelr 2495 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( Ws  A
)  e.  Ring )
83, 7jca 530 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( W  e.  Ring  /\  ( Ws  A
)  e.  Ring )
)
9 simpl3 1002 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  C_  V
)
10 simpl2 1001 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  .1.  e.  A )
119, 10jca 530 . . . 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 17747 . . . 4  |-  ( A  e.  (SubRing `  W
)  <->  ( ( W  e.  Ring  /\  ( Ws  A )  e.  Ring )  /\  ( A  C_  V  /\  .1.  e.  A
) ) )
158, 11, 14sylanbrc 662 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  (SubRing `  W ) )
16 assalmod 18286 . . . . 5  |-  ( S  e. AssAlg  ->  S  e.  LMod )
1716adantl 464 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  LMod )
18 assalmod 18286 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  LMod )
19 issubassa.l . . . . . 6  |-  L  =  ( LSubSp `  W )
204, 12, 19islss3 17923 . . . . 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 923 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  L )
2315, 22jca 530 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( A  e.  (SubRing `  W )  /\  A  e.  L
) )
2412subrgss 17748 . . . . . 6  |-  ( A  e.  (SubRing `  W
)  ->  A  C_  V
)
2524ad2antrl 726 . . . . 5  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  A  C_  V )
264, 12ressbas2 14897 . . . . 5  |-  ( A 
C_  V  ->  A  =  ( Base `  S
) )
2725, 26syl 17 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  A  =  ( Base `  S ) )
28 eqid 2402 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
294, 28resssca 14989 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  (Scalar `  W
)  =  (Scalar `  S ) )
3029ad2antrl 726 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  =  (Scalar `  S )
)
31 eqidd 2403 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
32 eqid 2402 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
334, 32ressvsca 14990 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .s `  W )  =  ( .s `  S ) )
3433ad2antrl 726 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .s `  W
)  =  ( .s
`  S ) )
35 eqid 2402 . . . . . 6  |-  ( .r
`  W )  =  ( .r `  W
)
364, 35ressmulr 14964 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  S ) )
3736ad2antrl 726 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .r `  W
)  =  ( .r
`  S ) )
38 simpr 459 . . . . 5  |-  ( ( A  e.  (SubRing `  W
)  /\  A  e.  L )  ->  A  e.  L )
394, 19lsslmod 17924 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  L )  ->  S  e.  LMod )
4018, 38, 39syl2an 475 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  LMod )
414subrgring 17750 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  S  e.  Ring )
4241ad2antrl 726 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  Ring )
4328assasca 18288 . . . . 5  |-  ( W  e. AssAlg  ->  (Scalar `  W )  e.  CRing )
4443adantr 463 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  e.  CRing )
45 simpll 752 . . . . 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 1003 . . . . 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 463 . . . . . 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 1004 . . . . . 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 3442 . . . . 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 1005 . . . . . 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 3442 . . . . 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 2402 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
5312, 28, 52, 32, 35assaass 18284 . . . . 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 1232 . . . 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 18285 . . . . 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 1232 . . . 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 18290 . . 3  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
58573ad2antl1 1159 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
5923, 58impbida 833 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3413   ` cfv 5568  (class class class)co 6277   Basecbs 14839   ↾s cress 14840   .rcmulr 14908  Scalarcsca 14910   .scvsca 14911   1rcur 17471   Ringcrg 17516   CRingccrg 17517  SubRingcsubrg 17743   LModclmod 17830   LSubSpclss 17896  AssAlgcasa 18276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-sbg 16381  df-subg 16520  df-mgp 17460  df-ur 17472  df-ring 17518  df-subrg 17745  df-lmod 17832  df-lss 17897  df-assa 18279
This theorem is referenced by:  mplassa  18434  ply1assa  18556
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