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Theorem sraassa 16339
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 2404 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 15824 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
54adantl 453 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  S  C_  ( Base `  W ) )
62, 5srabase 16205 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
72, 5srasca 16208 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
8 eqid 2404 . . . 4  |-  ( Ws  S )  =  ( Ws  S )
98subrgbas 15832 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
109adantl 453 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
112, 5sravsca 16209 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
122, 5sramulr 16207 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .r `  A ) )
131sralmod 16213 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1413adantl 453 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
15 crngrng 15629 . . . 4  |-  ( W  e.  CRing  ->  W  e.  Ring )
1615adantr 452 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  W  e.  Ring )
17 eqidd 2405 . . . 4  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  W )
)
182, 5sraaddg 16206 . . . . 5  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1918proplem3 13871 . . . 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 13871 . . . 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 15650 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( W  e.  Ring  <->  A  e.  Ring ) )
2216, 21mpbid 202 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  Ring )
238subrgcrng 15827 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e.  CRing )
2416adantr 452 . . 3  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  W  e.  Ring )
255adantr 452 . . . 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 963 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  x  e.  S )
2725, 26sseldd 3309 . . 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 964 . . 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 965 . . 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 2404 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
313, 30rngass 15635 . . 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 1186 . 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 2404 . . . . 5  |-  (mulGrp `  W )  =  (mulGrp `  W )
3433crngmgp 15627 . . . 4  |-  ( W  e.  CRing  ->  (mulGrp `  W
)  e. CMnd )
3534ad2antrr 707 . . 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 15609 . . . 4  |-  ( Base `  W )  =  (
Base `  (mulGrp `  W
) )
3733, 30mgpplusg 15607 . . . 4  |-  ( .r
`  W )  =  ( +g  `  (mulGrp `  W ) )
3836, 37cmn12 15387 . . 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 1186 . 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 16337 1  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. AssAlg )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3280   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   .rcmulr 13485  CMndccmn 15367  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616  SubRingcsubrg 15819   LModclmod 15905   subringAlg csra 16195  AssAlgcasa 16324
This theorem is referenced by:  rlmassa  16340
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mnd 14645  df-grp 14767  df-subg 14896  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-lmod 15907  df-sra 16199  df-assa 16327
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