Theorem srgass 13165

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 2177	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13162	. . . 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 1003	. . . 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 13147	. . . . 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 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 )
10	9, 7	eleqtrd 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 )
12	11, 7	eleqtrd 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 ) )
15	13, 14	mndass 12832	. . 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 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 )
18	1, 17	mgpplusgg 13145	. . . . 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 5895	. . 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 5895	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
22	21	oveq1d 5893	. . 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 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 ) )
24	19	oveqd 5895	. . 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 5895	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
26	25	oveq2d 5894	. . 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 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 ) ) )
28	16, 23, 27	3eqtr4d 2220	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 978   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5878   Basecbs 12465   +g cplusg 12539   .rcmulr 12540   Mndcmnd 12824  mulGrpcmgp 13141  SRingcsrg 13157

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-ltadd 7930

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-plusg 12552  df-mulr 12553  df-0g 12713  df-sgrp 12815  df-mnd 12825  df-mgp 13142  df-srg 13158

This theorem is referenced by:  srgpcomp  13184  srgpcompp  13185  srgpcomppsc  13186