Theorem rngass 13438

Description: Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)

Hypotheses

Ref	Expression
rngass.b	|- B = ( Base ` R )
rngass.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
rngass	|- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )

Proof of Theorem rngass

Step	Hyp	Ref	Expression
1		eqid 2193	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	rngmgp 13435	. . . 4 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
3	2	adantr 276	. . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> (mulGrp ` R ) e. Smgrp )
4		simpr1 1005	. . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
5		rngass.b	. . . . . 6 |- B = ( Base ` R )
6	1, 5	mgpbasg 13425	. . . . 5 |- ( R e. Rng -> B = ( Base ` (mulGrp ` R ) ) )
7	6	adantr 276	. . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> B = ( Base ` (mulGrp ` R ) ) )
8	4, 7	eleqtrd 2272	. . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. ( Base ` (mulGrp ` R ) ) )
9		simpr2 1006	. . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
10	9, 7	eleqtrd 2272	. . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11		simpr3 1007	. . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
12	11, 7	eleqtrd 2272	. . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13		eqid 2193	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14		eqid 2193	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
15	13, 14	sgrpass 12994	. . 3 |- ( ( (mulGrp ` R ) e. Smgrp /\ ( 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 1251	. 2 |- ( ( R e. Rng /\ ( 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		rngass.t	. . . . . . 7 |- .x. = ( .r ` R )
18	1, 17	mgpplusgg 13423	. . . . . 6 |- ( R e. Rng -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	oveqd 5936	. . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) )
20	18	oveqd 5936	. . . . . 6 |- ( R e. Rng -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
21	20	oveq1d 5934	. . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
22	19, 21	eqtrd 2226	. . . 4 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
23	18	oveqd 5936	. . . . 5 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) )
24	18	oveqd 5936	. . . . . 6 |- ( R e. Rng -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
25	24	oveq2d 5935	. . . . 5 |- ( R e. Rng -> ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
26	23, 25	eqtrd 2226	. . . 4 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
27	22, 26	eqeq12d 2208	. . 3 |- ( R e. Rng -> ( ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) <-> ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) ) )
28	27	adantr 276	. 2 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) <-> ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) ) )
29	16, 28	mpbird 167	1 |- ( ( R e. Rng /\ ( 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   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   .rcmulr 12699  Smgrpcsgrp 12987  mulGrpcmgp 13419  Rngcrng 13431

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-sgrp 12988  df-mgp 13420  df-rng 13432

This theorem is referenced by:  rngressid  13453  imasrng  13455  opprrng  13576  issubrng2  13709