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Theorem rngass 14083
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 2232 . . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
21rngmgp 14080 . . . 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 1030 . . . 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 14070 . . . . 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 2311 . . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. ( Base ` (mulGrp ` R ) ) )
9 simpr2 1031 . . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
109, 7eleqtrd 2311 . . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11 simpr3 1032 . . . 4 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
1211, 7eleqtrd 2311 . . 3 |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13 eqid 2232 . . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14 eqid 2232 . . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
1513, 14sgrpass 13621 . . 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 1276 . 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 14068 . . . . . 6 |- ( R e. Rng -> .x. = ( +g ` (mulGrp ` R ) ) )
1918oveqd 6067 . . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2018oveqd 6067 . . . . . 6 |- ( R e. Rng -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
2120oveq1d 6065 . . . . 5 |- ( R e. Rng -> ( ( X .x. Y ) ( +g ` (mulGrp ` R ) ) Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2219, 21eqtrd 2265 . . . 4 |- ( R e. Rng -> ( ( X .x. Y ) .x. Z ) = ( ( X ( +g ` (mulGrp ` R ) ) Y ) ( +g ` (mulGrp ` R ) ) Z ) )
2318oveqd 6067 . . . . 5 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) )
2418oveqd 6067 . . . . . 6 |- ( R e. Rng -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
2524oveq2d 6066 . . . . 5 |- ( R e. Rng -> ( X ( +g ` (mulGrp ` R ) ) ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
2623, 25eqtrd 2265 . . . 4 |- ( R e. Rng -> ( X .x. ( Y .x. Z ) ) = ( X ( +g ` (mulGrp ` R ) ) ( Y ( +g ` (mulGrp ` R ) ) Z ) ) )
2722, 26eqeq12d 2247 . . 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 1005   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291  Smgrpcsgrp 13614  mulGrpcmgp 14064  Rngcrng 14076
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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-sgrp 13615  df-mgp 14065  df-rng 14077
This theorem is referenced by:  rngressid  14098  imasrng  14100  opprrng  14221  issubrng2  14355
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