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Theorem ringass 13995
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 2229 . . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
21ringmgp 13981 . . . 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 1027 . . . 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 13905 . . . . 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 2308 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. ( Base ` (mulGrp ` R ) ) )
9 simpr2 1028 . . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
109, 7eleqtrd 2308 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11 simpr3 1029 . . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
1211, 7eleqtrd 2308 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13 eqid 2229 . . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14 eqid 2229 . . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
1513, 14mndass 13473 . . 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 1273 . 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 13903 . . . 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 6024 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
21 eqidd 2230 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z = Z )
2219, 20, 21oveq123d 6028 . 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 2230 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X = X )
2419oveqd 6024 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
2519, 23, 24oveq123d 6028 . 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 2272 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 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   Basecbs 13048   +g cplusg 13126   .rcmulr 13127   Mndcmnd 13465  mulGrpcmgp 13899   Ringcrg 13975
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-plusg 13139  df-mulr 13140  df-sgrp 13451  df-mnd 13466  df-mgp 13900  df-ring 13977
This theorem is referenced by:  ringinvnzdiv  14029  ringmneg1  14032  ringmneg2  14033  ringressid  14042  imasring  14043  opprring  14058  dvdsrtr  14081  dvdsrmul1  14082  unitgrp  14096  dvrass  14119  dvrcan1  14120  rdivmuldivd  14124  subrginv  14217  issubrg2  14221  unitrrg  14247  sralmod  14430
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