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Theorem rngass 13816
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 2207 . . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
21rngmgp 13813 . . . 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 1006 . . . 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 13803 . . . . 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 2286 . . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. ( Base ` (mulGrp ` R ) ) )
9 simpr2 1007 . . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
109, 7eleqtrd 2286 . . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11 simpr3 1008 . . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
1211, 7eleqtrd 2286 . . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13 eqid 2207 . . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14 eqid 2207 . . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
1513, 14sgrpass 13355 . . 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 1252 . 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 13801 . . . . . 6 |- ( R e. Rng -> .x. = ( +g ` (mulGrp ` R ) ) )
1918oveqd 5984 . . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2018oveqd 5984 . . . . . 6 |- ( R e. Rng -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
2120oveq1d 5982 . . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2219, 21eqtrd 2240 . . . 4 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2318oveqd 5984 . . . . 5 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) )
2418oveqd 5984 . . . . . 6 |- ( R e. Rng -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
2524oveq2d 5983 . . . . 5 |- ( R e. Rng -> ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
2623, 25eqtrd 2240 . . . 4 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
2722, 26eqeq12d 2222 . . 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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025  Smgrpcsgrp 13348  mulGrpcmgp 13797  Rngcrng 13809
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-sgrp 13349  df-mgp 13798  df-rng 13810
This theorem is referenced by:  rngressid  13831  imasrng  13833  opprrng  13954  issubrng2  14087
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