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Theorem sraassa 17374
Description: The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
Hypothesis
Ref Expression
sraassa.a  |-  A  =  ( (subringAlg  `  W ) `
 S )
Assertion
Ref Expression
sraassa  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. AssAlg )

Proof of Theorem sraassa
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sraassa.a . . . 4  |-  A  =  ( (subringAlg  `  W ) `
 S )
21a1i 11 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  =  ( (subringAlg  `  W ) `  S ) )
3 eqid 2441 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 16846 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
54adantl 463 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  S  C_  ( Base `  W ) )
62, 5srabase 17237 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
72, 5srasca 17240 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
8 eqid 2441 . . . 4  |-  ( Ws  S )  =  ( Ws  S )
98subrgbas 16854 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
109adantl 463 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
112, 5sravsca 17241 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
122, 5sramulr 17239 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .r `  A ) )
131sralmod 17246 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1413adantl 463 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
15 crngrng 16645 . . . 4  |-  ( W  e.  CRing  ->  W  e.  Ring )
1615adantr 462 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  W  e.  Ring )
17 eqidd 2442 . . . 4  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  W )
)
182, 5sraaddg 17238 . . . . 5  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1918proplem3 14625 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
) ) )  -> 
( x ( +g  `  W ) y )  =  ( x ( +g  `  A ) y ) )
2012proplem3 14625 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
) ) )  -> 
( x ( .r
`  W ) y )  =  ( x ( .r `  A
) y ) )
2117, 6, 19, 20rngpropd 16666 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( W  e.  Ring  <->  A  e.  Ring ) )
2216, 21mpbid 210 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  Ring )
238subrgcrng 16849 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e.  CRing )
2416adantr 462 . . 3  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  W  e.  Ring )
255adantr 462 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  S  C_  ( Base `  W
) )
26 simpr1 989 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  x  e.  S )
2725, 26sseldd 3354 . . 3  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  x  e.  ( Base `  W ) )
28 simpr2 990 . . 3  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  -> 
y  e.  ( Base `  W ) )
29 simpr3 991 . . 3  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  -> 
z  e.  ( Base `  W ) )
30 eqid 2441 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
313, 30rngass 16651 . . 3  |-  ( ( W  e.  Ring  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
)  /\  z  e.  ( Base `  W )
) )  ->  (
( x ( .r
`  W ) y ) ( .r `  W ) z )  =  ( x ( .r `  W ) ( y ( .r
`  W ) z ) ) )
3224, 27, 28, 29, 31syl13anc 1215 . 2  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  -> 
( ( x ( .r `  W ) y ) ( .r
`  W ) z )  =  ( x ( .r `  W
) ( y ( .r `  W ) z ) ) )
33 eqid 2441 . . . . 5  |-  (mulGrp `  W )  =  (mulGrp `  W )
3433crngmgp 16643 . . . 4  |-  ( W  e.  CRing  ->  (mulGrp `  W
)  e. CMnd )
3534ad2antrr 720 . . 3  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  -> 
(mulGrp `  W )  e. CMnd )
3633, 3mgpbas 16587 . . . 4  |-  ( Base `  W )  =  (
Base `  (mulGrp `  W
) )
3733, 30mgpplusg 16585 . . . 4  |-  ( .r
`  W )  =  ( +g  `  (mulGrp `  W ) )
3836, 37cmn12 16290 . . 3  |-  ( ( (mulGrp `  W )  e. CMnd  /\  ( y  e.  ( Base `  W
)  /\  x  e.  ( Base `  W )  /\  z  e.  ( Base `  W ) ) )  ->  ( y
( .r `  W
) ( x ( .r `  W ) z ) )  =  ( x ( .r
`  W ) ( y ( .r `  W ) z ) ) )
3935, 28, 27, 29, 38syl13anc 1215 . 2  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  -> 
( y ( .r
`  W ) ( x ( .r `  W ) z ) )  =  ( x ( .r `  W
) ( y ( .r `  W ) z ) ) )
406, 7, 10, 11, 12, 14, 22, 23, 32, 39isassad 17372 1  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. AssAlg )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    C_ wss 3325   ` cfv 5415  (class class class)co 6090   Basecbs 14170   ↾s cress 14171   +g cplusg 14234   .rcmulr 14235  CMndccmn 16270  mulGrpcmgp 16581   Ringcrg 16635   CRingccrg 16636  SubRingcsubrg 16841   LModclmod 16928  subringAlg csra 17227  AssAlgcasa 17359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-0g 14376  df-mnd 15411  df-grp 15538  df-subg 15671  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-subrg 16843  df-lmod 16930  df-sra 17231  df-assa 17362
This theorem is referenced by:  rlmassa  17375
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