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Theorem srgass 12967
Description: Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
srgcl.b  |-  B  =  ( Base `  R
)
srgcl.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
srgass  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( X  .x.  ( Y 
.x.  Z ) ) )

Proof of Theorem srgass
StepHypRef Expression
1 eqid 2177 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
21srgmgp 12964 . . . 4  |-  ( R  e. SRing  ->  (mulGrp `  R )  e.  Mnd )
32adantr 276 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  (mulGrp `  R
)  e.  Mnd )
4 simpr1 1003 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  B )
5 srgcl.b . . . . . 6  |-  B  =  ( Base `  R
)
61, 5mgpbasg 12950 . . . . 5  |-  ( R  e. SRing  ->  B  =  (
Base `  (mulGrp `  R
) ) )
76adantr 276 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  B  =  ( Base `  (mulGrp `  R
) ) )
84, 7eleqtrd 2256 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  ( Base `  (mulGrp `  R
) ) )
9 simpr2 1004 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  B )
109, 7eleqtrd 2256 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  ( Base `  (mulGrp `  R
) ) )
11 simpr3 1005 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  B )
1211, 7eleqtrd 2256 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  ( Base `  (mulGrp `  R
) ) )
13 eqid 2177 . . . 4  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
14 eqid 2177 . . . 4  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
1513, 14mndass 12704 . . 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 ) ) )
163, 8, 10, 12, 15syl13anc 1240 . 2  |-  ( ( R  e. SRing  /\  ( 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 srgcl.t . . . . . 6  |-  .x.  =  ( .r `  R )
181, 17mgpplusgg 12948 . . . . 5  |-  ( R  e. SRing  ->  .x.  =  ( +g  `  (mulGrp `  R
) ) )
1918adantr 276 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  .x.  =  ( +g  `  (mulGrp `  R ) ) )
2019oveqd 5885 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( ( X  .x.  Y
) ( +g  `  (mulGrp `  R ) ) Z ) )
2119oveqd 5885 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  Y )  =  ( X ( +g  `  (mulGrp `  R ) ) Y ) )
2221oveq1d 5883 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  Y ) ( +g  `  (mulGrp `  R ) ) Z )  =  ( ( X ( +g  `  (mulGrp `  R ) ) Y ) ( +g  `  (mulGrp `  R ) ) Z ) )
2320, 22eqtrd 2210 . 2  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( ( X ( +g  `  (mulGrp `  R )
) Y ) ( +g  `  (mulGrp `  R ) ) Z ) )
2419oveqd 5885 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  ( Y  .x.  Z
) )  =  ( X ( +g  `  (mulGrp `  R ) ) ( Y  .x.  Z ) ) )
2519oveqd 5885 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .x.  Z )  =  ( Y ( +g  `  (mulGrp `  R ) ) Z ) )
2625oveq2d 5884 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X
( +g  `  (mulGrp `  R ) ) ( Y  .x.  Z ) )  =  ( X ( +g  `  (mulGrp `  R ) ) ( Y ( +g  `  (mulGrp `  R ) ) Z ) ) )
2724, 26eqtrd 2210 . 2  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  ( Y  .x.  Z
) )  =  ( X ( +g  `  (mulGrp `  R ) ) ( Y ( +g  `  (mulGrp `  R ) ) Z ) ) )
2816, 23, 273eqtr4d 2220 1  |-  ( ( R  e. SRing  /\  ( 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    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   .rcmulr 12506   Mndcmnd 12696  mulGrpcmgp 12944  SRingcsrg 12959
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-sgrp 12687  df-mnd 12697  df-mgp 12945  df-srg 12960
This theorem is referenced by:  srgpcomp  12986  srgpcompp  12987  srgpcomppsc  12988
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