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Theorem rngass 13571
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
StepHypRef Expression
1 eqid 2196 . . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
21rngmgp 13568 . . . 4 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
32adantr 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 )
61, 5mgpbasg 13558 . . . . 5 |- ( R e. Rng -> B = ( Base ` (mulGrp ` R ) ) )
76adantr 276 . . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> B = ( Base ` (mulGrp ` R ) ) )
84, 7eleqtrd 2275 . . 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 )
109, 7eleqtrd 2275 . . 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 )
1211, 7eleqtrd 2275 . . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13 eqid 2196 . . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14 eqid 2196 . . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
1513, 14sgrpass 13110 . . 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 ) ) )
163, 8, 10, 12, 15syl13anc 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 )
181, 17mgpplusgg 13556 . . . . . 6 |- ( R e. Rng -> .x. = ( +g ` (mulGrp ` R ) ) )
1918oveqd 5942 . . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2018oveqd 5942 . . . . . 6 |- ( R e. Rng -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
2120oveq1d 5940 . . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2219, 21eqtrd 2229 . . . 4 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2318oveqd 5942 . . . . 5 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) )
2418oveqd 5942 . . . . . 6 |- ( R e. Rng -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
2524oveq2d 5941 . . . . 5 |- ( R e. Rng -> ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
2623, 25eqtrd 2229 . . . 4 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
2722, 26eqeq12d 2211 . . 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 ) ) ) )
2827adantr 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 ) ) ) )
2916, 28mpbird 167 1 |- ( ( R e. Rng /\ ( 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   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   .rcmulr 12781  Smgrpcsgrp 13103  mulGrpcmgp 13552  Rngcrng 13564
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-plusg 12793  df-mulr 12794  df-sgrp 13104  df-mgp 13553  df-rng 13565
This theorem is referenced by:  rngressid  13586  imasrng  13588  opprrng  13709  issubrng2  13842
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