Theorem srgass 13894

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 2207	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13891	. . . 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 1006	. . . 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 13849	. . . . 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 2286	. . 3 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. ( Base ` (mulGrp ` R ) ) )
9		simpr2 1007	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
10	9, 7	eleqtrd 2286	. . 3 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11		simpr3 1008	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
12	11, 7	eleqtrd 2286	. . 3 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. ( Base ` (mulGrp ` R ) ) )
13		eqid 2207	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14		eqid 2207	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
15	13, 14	mndass 13417	. . 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 1252	. 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 13847	. . . . 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 5986	. . 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 5986	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
22	21	oveq1d 5984	. . 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 2240	. 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 5986	. . 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 5986	. . . 4 |- ( ( R e. SRing /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .x. Z ) = ( Y ( +g ` (mulGrp ` R ) ) Z ) )
26	25	oveq2d 5985	. . 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 2240	. 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 2250	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 981   = wceq 1373   e. wcel 2178   ` cfv 5291  (class class class)co 5969   Basecbs 12993   +g cplusg 13070   .rcmulr 13071   Mndcmnd 13409  mulGrpcmgp 13843  SRingcsrg 13886

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-addcom 8062  ax-addass 8064  ax-i2m1 8067  ax-0lt1 8068  ax-0id 8070  ax-rnegex 8071  ax-pre-ltirr 8074  ax-pre-ltadd 8078

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-iota 5252  df-fun 5293  df-fn 5294  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-pnf 8146  df-mnf 8147  df-ltxr 8149  df-inn 9074  df-2 9132  df-3 9133  df-ndx 12996  df-slot 12997  df-base 12999  df-sets 13000  df-plusg 13083  df-mulr 13084  df-0g 13251  df-sgrp 13395  df-mnd 13410  df-mgp 13844  df-srg 13887

This theorem is referenced by:  srgpcomp  13913  srgpcompp  13914  srgpcomppsc  13915