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Theorem ringass 14294
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 2238 . . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
21ringmgp 14280 . . . 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 1034 . . . 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 14200 . . . . 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 2317 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. ( Base ` (mulGrp ` R ) ) )
9 simpr2 1035 . . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
109, 7eleqtrd 2317 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11 simpr3 1036 . . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
1211, 7eleqtrd 2317 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13 eqid 2238 . . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14 eqid 2238 . . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
1513, 14mndass 13714 . . 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 1280 . 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 14198 . . . 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 6092 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
21 eqidd 2239 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z = Z )
2219, 20, 21oveq123d 6096 . 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 2239 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X = X )
2419oveqd 6092 . . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
2519, 23, 24oveq123d 6096 . 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 2281 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 1009   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   .rcmulr 13409   Mndcmnd 13706  mulGrpcmgp 14194   Ringcrg 14274
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285
This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-sgrp 13694  df-mnd 13707  df-mgp 14195  df-ring 14276
This theorem is referenced by:  ringinvnzdiv  14328  ringmneg1  14331  ringmneg2  14332  ringressid  14341  imasring  14342  opprring  14357  dvdsrtr  14381  dvdsrmul1  14382  unitgrp  14396  dvrass  14419  dvrcan1  14420  rdivmuldivd  14424  subrginv  14518  issubrg2  14522  unitrrg  14549  sralmod  14759
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