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Theorem sraassa 17529
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 2454 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 16999 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
54adantl 466 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  S  C_  ( Base `  W ) )
62, 5srabase 17392 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
72, 5srasca 17395 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
8 eqid 2454 . . . 4  |-  ( Ws  S )  =  ( Ws  S )
98subrgbas 17007 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
109adantl 466 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
112, 5sravsca 17396 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
122, 5sramulr 17394 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .r `  A ) )
131sralmod 17401 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1413adantl 466 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
15 crngrng 16788 . . . 4  |-  ( W  e.  CRing  ->  W  e.  Ring )
1615adantr 465 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  W  e.  Ring )
17 eqidd 2455 . . . 4  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  W )
)
182, 5sraaddg 17393 . . . . 5  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1918proplem3 14752 . . . 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 14752 . . . 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 16809 . . 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 17002 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e.  CRing )
2416adantr 465 . . 3  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  W  e.  Ring )
255adantr 465 . . . 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 994 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  x  e.  S )
2725, 26sseldd 3468 . . 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 995 . . 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 996 . . 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 2454 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
313, 30rngass 16794 . . 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 1221 . 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 2454 . . . . 5  |-  (mulGrp `  W )  =  (mulGrp `  W )
3433crngmgp 16786 . . . 4  |-  ( W  e.  CRing  ->  (mulGrp `  W
)  e. CMnd )
3534ad2antrr 725 . . 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 16729 . . . 4  |-  ( Base `  W )  =  (
Base `  (mulGrp `  W
) )
3733, 30mgpplusg 16727 . . . 4  |-  ( .r
`  W )  =  ( +g  `  (mulGrp `  W ) )
3836, 37cmn12 16422 . . 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 1221 . 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 17527 1  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. AssAlg )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3439   ` cfv 5529  (class class class)co 6203   Basecbs 14296   ↾s cress 14297   +g cplusg 14361   .rcmulr 14362  CMndccmn 16402  mulGrpcmgp 16723   Ringcrg 16778   CRingccrg 16779  SubRingcsubrg 16994   LModclmod 17081  subringAlg csra 17382  AssAlgcasa 17514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-ip 14379  df-0g 14503  df-mnd 15538  df-grp 15668  df-subg 15801  df-cmn 16404  df-mgp 16724  df-ur 16736  df-rng 16780  df-cring 16781  df-subrg 16996  df-lmod 17083  df-sra 17386  df-assa 17517
This theorem is referenced by:  rlmassa  17530
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