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Theorem issubassa 16346
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 960 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  W  e. AssAlg )
2 assarng 16343 . . . . . 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 assarng 16343 . . . . . . 7  |-  ( S  e. AssAlg  ->  S  e.  Ring )
65adantl 453 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  Ring )
74, 6syl5eqelr 2497 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( Ws  A
)  e.  Ring )
83, 7jca 519 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( W  e.  Ring  /\  ( Ws  A
)  e.  Ring )
)
9 simpl3 962 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  C_  V
)
10 simpl2 961 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  .1.  e.  A )
119, 10jca 519 . . . 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 15831 . . . 4  |-  ( A  e.  (SubRing `  W
)  <->  ( ( W  e.  Ring  /\  ( Ws  A )  e.  Ring )  /\  ( A  C_  V  /\  .1.  e.  A
) ) )
158, 11, 14sylanbrc 646 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  (SubRing `  W ) )
16 assalmod 16342 . . . . 5  |-  ( S  e. AssAlg  ->  S  e.  LMod )
1716adantl 453 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  LMod )
18 assalmod 16342 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  LMod )
19 issubassa.l . . . . . 6  |-  L  =  ( LSubSp `  W )
204, 12, 19islss3 15998 . . . . 5  |-  ( W  e.  LMod  ->  ( A  e.  L  <->  ( A  C_  V  /\  S  e. 
LMod ) ) )
211, 18, 203syl 19 . . . 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 889 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  L )
2315, 22jca 519 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( A  e.  (SubRing `  W )  /\  A  e.  L
) )
2412subrgss 15832 . . . . . 6  |-  ( A  e.  (SubRing `  W
)  ->  A  C_  V
)
2524ad2antrl 709 . . . . 5  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  A  C_  V )
264, 12ressbas2 13483 . . . . 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 2412 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
294, 28resssca 13567 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  (Scalar `  W
)  =  (Scalar `  S ) )
3029ad2antrl 709 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  =  (Scalar `  S )
)
31 eqidd 2413 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
32 eqid 2412 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
334, 32ressvsca 13568 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .s `  W )  =  ( .s `  S ) )
3433ad2antrl 709 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .s `  W
)  =  ( .s
`  S ) )
35 eqid 2412 . . . . . 6  |-  ( .r
`  W )  =  ( .r `  W
)
364, 35ressmulr 13545 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  S ) )
3736ad2antrl 709 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .r `  W
)  =  ( .r
`  S ) )
38 simpr 448 . . . . 5  |-  ( ( A  e.  (SubRing `  W
)  /\  A  e.  L )  ->  A  e.  L )
394, 19lsslmod 15999 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  L )  ->  S  e.  LMod )
4018, 38, 39syl2an 464 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  LMod )
414subrgrng 15834 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  S  e.  Ring )
4241ad2antrl 709 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  Ring )
4328assasca 16344 . . . . 5  |-  ( W  e. AssAlg  ->  (Scalar `  W )  e.  CRing )
4443adantr 452 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  e.  CRing )
45 simpll 731 . . . . 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 963 . . . . 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 452 . . . . . 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 964 . . . . . 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 3317 . . . . 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 965 . . . . . 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 3317 . . . . 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 2412 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
5312, 28, 52, 32, 35assaass 16340 . . . . 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 1186 . . . 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 16341 . . . . 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 1186 . . . 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 16345 . . 3  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
58573ad2antl1 1119 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
5923, 58impbida 806 1  |-  ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V
)  ->  ( S  e. AssAlg  <-> 
( A  e.  (SubRing `  W )  /\  A  e.  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3288   ` cfv 5421  (class class class)co 6048   Basecbs 13432   ↾s cress 13433   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496   Ringcrg 15623   CRingccrg 15624   1rcur 15625  SubRingcsubrg 15827   LModclmod 15913   LSubSpclss 15971  AssAlgcasa 16332
This theorem is referenced by:  mplassa  16480  ply1assa  16560
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-0g 13690  df-mnd 14653  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-mgp 15612  df-rng 15626  df-ur 15628  df-subrg 15829  df-lmod 15915  df-lss 15972  df-assa 16335
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