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Theorem sraassa 18101
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 2457 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 17557 . . . 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 17951 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
72, 5srasca 17954 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
8 eqid 2457 . . . 4  |-  ( Ws  S )  =  ( Ws  S )
98subrgbas 17565 . . 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 17955 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
122, 5sramulr 17953 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .r `  A ) )
131sralmod 17960 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1413adantl 466 . 2  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
15 crngring 17336 . . . 4  |-  ( W  e.  CRing  ->  W  e.  Ring )
1615adantr 465 . . 3  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  W  e.  Ring )
17 eqidd 2458 . . . 4  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  W )
)
182, 5sraaddg 17952 . . . . 5  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1918oveqdr 6320 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
) ) )  -> 
( x ( +g  `  W ) y )  =  ( x ( +g  `  A ) y ) )
2012oveqdr 6320 . . . 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, 20ringpropd 17357 . . 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 17560 . 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 1002 . . . 4  |-  ( ( ( W  e.  CRing  /\  S  e.  (SubRing `  W
) )  /\  (
x  e.  S  /\  y  e.  ( Base `  W )  /\  z  e.  ( Base `  W
) ) )  ->  x  e.  S )
2725, 26sseldd 3500 . . 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 1003 . . 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 1004 . . 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 2457 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
313, 30ringass 17342 . . 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 1230 . 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 2457 . . . . 5  |-  (mulGrp `  W )  =  (mulGrp `  W )
3433crngmgp 17333 . . . 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 17274 . . . 4  |-  ( Base `  W )  =  (
Base `  (mulGrp `  W
) )
3733, 30mgpplusg 17272 . . . 4  |-  ( .r
`  W )  =  ( +g  `  (mulGrp `  W ) )
3836, 37cmn12 16945 . . 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 1230 . 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 18099 1  |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. AssAlg )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14644   ↾s cress 14645   +g cplusg 14712   .rcmulr 14713  CMndccmn 16925  mulGrpcmgp 17268   Ringcrg 17325   CRingccrg 17326  SubRingcsubrg 17552   LModclmod 17639  subringAlg csra 17941  AssAlgcasa 18085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-subg 16325  df-cmn 16927  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-subrg 17554  df-lmod 17641  df-sra 17945  df-assa 18088
This theorem is referenced by:  rlmassa  18102
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