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Theorem srgass 13342
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 2189 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
21srgmgp 13339 . . . 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 1005 . . . 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 13297 . . . . 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 2268 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  X  e.  ( Base `  (mulGrp `  R
) ) )
9 simpr2 1006 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  B )
109, 7eleqtrd 2268 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Y  e.  ( Base `  (mulGrp `  R
) ) )
11 simpr3 1007 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  B )
1211, 7eleqtrd 2268 . . 3  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  e.  ( Base `  (mulGrp `  R
) ) )
13 eqid 2189 . . . 4  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
14 eqid 2189 . . . 4  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
1513, 14mndass 12900 . . 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 1251 . 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 13295 . . . . 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 5914 . . 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 5914 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  Y )  =  ( X ( +g  `  (mulGrp `  R ) ) Y ) )
2221oveq1d 5912 . . 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 2222 . 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 5914 . . 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 5914 . . . 4  |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .x.  Z )  =  ( Y ( +g  `  (mulGrp `  R ) ) Z ) )
2625oveq2d 5913 . . 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 2222 . 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 2232 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 980    = wceq 1364    e. wcel 2160   ` cfv 5235  (class class class)co 5897   Basecbs 12515   +g cplusg 12592   .rcmulr 12593   Mndcmnd 12892  mulGrpcmgp 13291  SRingcsrg 13334
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-pre-ltirr 7954  ax-pre-ltadd 7958
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-ltxr 8028  df-inn 8951  df-2 9009  df-3 9010  df-ndx 12518  df-slot 12519  df-base 12521  df-sets 12522  df-plusg 12605  df-mulr 12606  df-0g 12766  df-sgrp 12880  df-mnd 12893  df-mgp 13292  df-srg 13335
This theorem is referenced by:  srgpcomp  13361  srgpcompp  13362  srgpcomppsc  13363
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