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Theorem ringass 13204
Description: Associative law for multiplication in a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ringcl.b  |-  B  =  ( Base `  R
)
ringcl.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
ringass  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( X  .x.  ( Y 
.x.  Z ) ) )

Proof of Theorem ringass
StepHypRef Expression
1 eqid 2177 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
21ringmgp 13190 . . . 4  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
32adantr 276 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  (mulGrp `  R
)  e.  Mnd )
4 simpr1 1003 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  B )
5 ringcl.b . . . . . 6  |-  B  =  ( Base `  R
)
61, 5mgpbasg 13141 . . . . 5  |-  ( R  e.  Ring  ->  B  =  ( Base `  (mulGrp `  R ) ) )
76adantr 276 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  B  =  ( Base `  (mulGrp `  R
) ) )
84, 7eleqtrd 2256 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  ( Base `  (mulGrp `  R
) ) )
9 simpr2 1004 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  B )
109, 7eleqtrd 2256 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  ( Base `  (mulGrp `  R
) ) )
11 simpr3 1005 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  B )
1211, 7eleqtrd 2256 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  ( Base `  (mulGrp `  R
) ) )
13 eqid 2177 . . . 4  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
14 eqid 2177 . . . 4  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
1513, 14mndass 12830 . . 3  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  ( X  e.  ( Base `  (mulGrp `  R ) )  /\  Y  e.  ( Base `  (mulGrp `  R )
)  /\  Z  e.  ( Base `  (mulGrp `  R
) ) ) )  ->  ( ( X ( +g  `  (mulGrp `  R ) ) Y ) ( +g  `  (mulGrp `  R ) ) Z )  =  ( X ( +g  `  (mulGrp `  R ) ) ( Y ( +g  `  (mulGrp `  R ) ) Z ) ) )
163, 8, 10, 12, 15syl13anc 1240 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X ( +g  `  (mulGrp `  R ) ) Y ) ( +g  `  (mulGrp `  R ) ) Z )  =  ( X ( +g  `  (mulGrp `  R ) ) ( Y ( +g  `  (mulGrp `  R ) ) Z ) ) )
17 ringcl.t . . . . 5  |-  .x.  =  ( .r `  R )
181, 17mgpplusgg 13139 . . . 4  |-  ( R  e.  Ring  ->  .x.  =  ( +g  `  (mulGrp `  R ) ) )
1918adantr 276 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  .x.  =  ( +g  `  (mulGrp `  R ) ) )
2019oveqd 5894 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  Y )  =  ( X ( +g  `  (mulGrp `  R ) ) Y ) )
21 eqidd 2178 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  =  Z )
2219, 20, 21oveq123d 5898 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( ( X ( +g  `  (mulGrp `  R )
) Y ) ( +g  `  (mulGrp `  R ) ) Z ) )
23 eqidd 2178 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  =  X )
2419oveqd 5894 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .x.  Z )  =  ( Y ( +g  `  (mulGrp `  R ) ) Z ) )
2519, 23, 24oveq123d 5898 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  ( Y  .x.  Z
) )  =  ( X ( +g  `  (mulGrp `  R ) ) ( Y ( +g  `  (mulGrp `  R ) ) Z ) ) )
2616, 22, 253eqtr4d 2220 1  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( X  .x.  ( Y 
.x.  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   .rcmulr 12539   Mndcmnd 12822  mulGrpcmgp 13135   Ringcrg 13184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-sgrp 12813  df-mnd 12823  df-mgp 13136  df-ring 13186
This theorem is referenced by:  ringinvnzdiv  13232  ringmneg1  13235  ringmneg2  13236  ringressid  13243  opprring  13254  dvdsrtr  13275  dvdsrmul1  13276  unitgrp  13290  dvrass  13313  dvrcan1  13314  rdivmuldivd  13318  subrginv  13363  issubrg2  13367
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