Theorem ringass 13152

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

Step	Hyp	Ref	Expression
1		eqid 2177	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	ringmgp 13138	. . . 4 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
3	2	adantr 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 )
6	1, 5	mgpbasg 13089	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
7	6	adantr 276	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> B = ( Base ` (mulGrp ` R ) ) )
8	4, 7	eleqtrd 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 )
10	9, 7	eleqtrd 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 )
12	11, 7	eleqtrd 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 ) )
15	13, 14	mndass 12779	. . 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 ) ) )
16	3, 8, 10, 12, 15	syl13anc 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 )
18	1, 17	mgpplusgg 13087	. . . 4 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	adantr 276	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> .x. = ( +g ` (mulGrp ` R ) ) )
20	19	oveqd 5891	. . 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 )
22	19, 20, 21	oveq123d 5895	. 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 )
24	19	oveqd 5891	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
25	19, 23, 24	oveq123d 5895	. 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 ) ) )
26	16, 22, 25	3eqtr4d 2220	1 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   ` cfv 5216  (class class class)co 5874   Basecbs 12456   +g cplusg 12530   .rcmulr 12531   Mndcmnd 12771  mulGrpcmgp 13083   Ringcrg 13132

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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-sgrp 12762  df-mnd 12772  df-mgp 13084  df-ring 13134

This theorem is referenced by:  ringinvnzdiv  13180  ringmneg1  13183  ringmneg2  13184  ringressid  13191  opprring  13202  dvdsrtr  13223  dvdsrmul1  13224  unitgrp  13238  dvrass  13261  dvrcan1  13262  rdivmuldivd  13266  subrginv  13318  issubrg2  13322