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

Step	Hyp	Ref	Expression
1		eqid 2189	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13339	. . . 4 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3	2	adantr 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 )
6	1, 5	mgpbasg 13297	. . . . 5 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
7	6	adantr 276	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> B = ( Base ` (mulGrp ` R ) ) )
8	4, 7	eleqtrd 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 )
10	9, 7	eleqtrd 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 )
12	11, 7	eleqtrd 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 ) )
15	13, 14	mndass 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 ) ) )
16	3, 8, 10, 12, 15	syl13anc 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 )
18	1, 17	mgpplusgg 13295	. . . . 5 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	adantr 276	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> .x. = ( +g ` (mulGrp ` R ) ) )
20	19	oveqd 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 ) )
21	19	oveqd 5914	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
22	21	oveq1d 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 ) )
23	20, 22	eqtrd 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 ) )
24	19	oveqd 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 ) ) )
25	19	oveqd 5914	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
26	25	oveq2d 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 ) ) )
27	24, 26	eqtrd 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 ) ) )
28	16, 23, 27	3eqtr4d 2232	1 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )

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