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Theorem issubassa 16064
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 958 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  W  e. AssAlg )
2 assarng 16061 . . . . . 6  |-  ( W  e. AssAlg  ->  W  e.  Ring )
31, 2syl 15 . . . . 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 16061 . . . . . . 7  |-  ( S  e. AssAlg  ->  S  e.  Ring )
65adantl 452 . . . . . 6  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  Ring )
74, 6syl5eqelr 2368 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( Ws  A
)  e.  Ring )
83, 7jca 518 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( W  e.  Ring  /\  ( Ws  A
)  e.  Ring )
)
9 simpl3 960 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  C_  V
)
10 simpl2 959 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  .1.  e.  A )
119, 10jca 518 . . . 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 15545 . . . 4  |-  ( A  e.  (SubRing `  W
)  <->  ( ( W  e.  Ring  /\  ( Ws  A )  e.  Ring )  /\  ( A  C_  V  /\  .1.  e.  A
) ) )
158, 11, 14sylanbrc 645 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  (SubRing `  W ) )
16 assalmod 16060 . . . . 5  |-  ( S  e. AssAlg  ->  S  e.  LMod )
1716adantl 452 . . . 4  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  S  e.  LMod )
18 assalmod 16060 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  LMod )
19 issubassa.l . . . . . 6  |-  L  =  ( LSubSp `  W )
204, 12, 19islss3 15716 . . . . 5  |-  ( W  e.  LMod  ->  ( A  e.  L  <->  ( A  C_  V  /\  S  e. 
LMod ) ) )
211, 18, 203syl 18 . . . 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 888 . . 3  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  A  e.  L )
2315, 22jca 518 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  S  e. AssAlg )  ->  ( A  e.  (SubRing `  W )  /\  A  e.  L
) )
2412subrgss 15546 . . . . . 6  |-  ( A  e.  (SubRing `  W
)  ->  A  C_  V
)
2524ad2antrl 708 . . . . 5  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  A  C_  V )
264, 12ressbas2 13199 . . . . 5  |-  ( A 
C_  V  ->  A  =  ( Base `  S
) )
2725, 26syl 15 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  A  =  ( Base `  S ) )
28 eqid 2283 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
294, 28resssca 13283 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  (Scalar `  W
)  =  (Scalar `  S ) )
3029ad2antrl 708 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  =  (Scalar `  S )
)
31 eqidd 2284 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
32 eqid 2283 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
334, 32ressvsca 13284 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .s `  W )  =  ( .s `  S ) )
3433ad2antrl 708 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .s `  W
)  =  ( .s
`  S ) )
35 eqid 2283 . . . . . 6  |-  ( .r
`  W )  =  ( .r `  W
)
364, 35ressmulr 13261 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  S ) )
3736ad2antrl 708 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( .r `  W
)  =  ( .r
`  S ) )
38 simpr 447 . . . . 5  |-  ( ( A  e.  (SubRing `  W
)  /\  A  e.  L )  ->  A  e.  L )
394, 19lsslmod 15717 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  L )  ->  S  e.  LMod )
4018, 38, 39syl2an 463 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  LMod )
414subrgrng 15548 . . . . 5  |-  ( A  e.  (SubRing `  W
)  ->  S  e.  Ring )
4241ad2antrl 708 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e.  Ring )
4328assasca 16062 . . . . 5  |-  ( W  e. AssAlg  ->  (Scalar `  W )  e.  CRing )
4443adantr 451 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
(Scalar `  W )  e.  CRing )
45 simpll 730 . . . . 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 961 . . . . 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 451 . . . . . 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 962 . . . . . 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 3181 . . . . 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 963 . . . . . 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 3181 . . . . 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 2283 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
5312, 28, 52, 32, 35assaass 16058 . . . . 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 1184 . . . 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 16059 . . . . 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 1184 . . . 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 16063 . . 3  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
58573ad2antl1 1117 . 2  |-  ( ( ( W  e. AssAlg  /\  .1.  e.  A  /\  A  C_  V )  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  ->  S  e. AssAlg )
5923, 58impbida 805 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   Ringcrg 15337   CRingccrg 15338   1rcur 15339  SubRingcsubrg 15541   LModclmod 15627   LSubSpclss 15689  AssAlgcasa 16050
This theorem is referenced by:  mplassa  16198  ply1assa  16278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-assa 16053
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