Theorem ringass 14028

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 2231	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	ringmgp 14014	. . . 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 1029	. . . 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 13938	. . . . 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 2310	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. ( Base ` (mulGrp ` R ) ) )
9		simpr2 1030	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
10	9, 7	eleqtrd 2310	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11		simpr3 1031	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
12	11, 7	eleqtrd 2310	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13		eqid 2231	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14		eqid 2231	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
15	13, 14	mndass 13506	. . 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 1275	. 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 13936	. . . 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 6034	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
21		eqidd 2232	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z = Z )
22	19, 20, 21	oveq123d 6038	. 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 2232	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X = X )
24	19	oveqd 6034	. . 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 6038	. 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 2274	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 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   Mndcmnd 13498  mulGrpcmgp 13932   Ringcrg 14008

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-sgrp 13484  df-mnd 13499  df-mgp 13933  df-ring 14010

This theorem is referenced by:  ringinvnzdiv  14062  ringmneg1  14065  ringmneg2  14066  ringressid  14075  imasring  14076  opprring  14091  dvdsrtr  14114  dvdsrmul1  14115  unitgrp  14129  dvrass  14152  dvrcan1  14153  rdivmuldivd  14157  subrginv  14250  issubrg2  14254  unitrrg  14280  sralmod  14463