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Theorem issubassa 16163
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 16160 . . . . . 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 16160 . . . . . . 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 2443 . . . . 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 15644 . . . 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 16159 . . . . 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 16159 . . . . 5  |-  ( W  e. AssAlg  ->  W  e.  LMod )
19 issubassa.l . . . . . 6  |-  L  =  ( LSubSp `  W )
204, 12, 19islss3 15815 . . . . 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 15645 . . . . . 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 13296 . . . . 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 2358 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
294, 28resssca 13380 . . . . 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 2359 . . . 4  |-  ( ( W  e. AssAlg  /\  ( A  e.  (SubRing `  W
)  /\  A  e.  L ) )  -> 
( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
32 eqid 2358 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
334, 32ressvsca 13381 . . . . 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 2358 . . . . . 6  |-  ( .r
`  W )  =  ( .r `  W
)
364, 35ressmulr 13358 . . . . 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 15816 . . . . 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 15647 . . . . 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 16161 . . . . 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 3257 . . . . 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 3257 . . . . 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 2358 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
5312, 28, 52, 32, 35assaass 16157 . . . . 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 16158 . . . . 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 16162 . . 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 1642    e. wcel 1710    C_ wss 3228   ` cfv 5337  (class class class)co 5945   Basecbs 13245   ↾s cress 13246   .rcmulr 13306  Scalarcsca 13308   .scvsca 13309   Ringcrg 15436   CRingccrg 15437   1rcur 15438  SubRingcsubrg 15640   LModclmod 15726   LSubSpclss 15788  AssAlgcasa 16149
This theorem is referenced by:  mplassa  16297  ply1assa  16377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-subg 14717  df-mgp 15425  df-rng 15439  df-ur 15441  df-subrg 15642  df-lmod 15728  df-lss 15789  df-assa 16152
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