Theorem mulgass3 13641

Description: An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
mulgass3.b	|- B = ( Base ` R )
mulgass3.m	|- .x. = (.g ` R )
mulgass3.t	|- .X. = ( .r ` R )

Assertion

Ref	Expression
mulgass3	|- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( X .X. ( N .x. Y ) ) = ( N .x. ( X .X. Y ) ) )

Proof of Theorem mulgass3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2196	. . . . . 6 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13635	. . . . 5 |- ( R e. Ring -> (oppr ` R ) e. Ring )
3	2	adantr 276	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> (oppr ` R ) e. Ring )
4		simpr1 1005	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> N e. ZZ )
5		simpr3 1007	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> Y e. B )
6		mulgass3.b	. . . . . . 7 |- B = ( Base ` R )
7	1, 6	opprbasg 13631	. . . . . 6 |- ( R e. Ring -> B = ( Base ` (oppr ` R ) ) )
8	7	adantr 276	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> B = ( Base ` (oppr ` R ) ) )
9	5, 8	eleqtrd 2275	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> Y e. ( Base ` (oppr ` R ) ) )
10		simpr2 1006	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> X e. B )
11	10, 8	eleqtrd 2275	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> X e. ( Base ` (oppr ` R ) ) )
12		eqid 2196	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
13		eqid 2196	. . . . 5 |- (.g ` (oppr ` R ) ) = (.g ` (oppr ` R ) )
14		eqid 2196	. . . . 5 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
15	12, 13, 14	mulgass2 13614	. . . 4 |- ( ( (oppr ` R ) e. Ring /\ ( N e. ZZ /\ Y e. ( Base ` (oppr ` R ) ) /\ X e. ( Base ` (oppr ` R ) ) ) ) -> ( ( N (.g ` (oppr ` R ) ) Y ) ( .r ` (oppr ` R ) ) X ) = ( N (.g ` (oppr ` R ) ) ( Y ( .r ` (oppr ` R ) ) X ) ) )
16	3, 4, 9, 11, 15	syl13anc 1251	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( ( N (.g ` (oppr ` R ) ) Y ) ( .r ` (oppr ` R ) ) X ) = ( N (.g ` (oppr ` R ) ) ( Y ( .r ` (oppr ` R ) ) X ) ) )
17		simpl 109	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> R e. Ring )
18	3	ringgrpd 13561	. . . . . 6 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> (oppr ` R ) e. Grp )
19	12, 13, 18, 4, 9	mulgcld 13274	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N (.g ` (oppr ` R ) ) Y ) e. ( Base ` (oppr ` R ) ) )
20	19, 8	eleqtrrd 2276	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N (.g ` (oppr ` R ) ) Y ) e. B )
21		mulgass3.t	. . . . 5 |- .X. = ( .r ` R )
22	6, 21, 1, 14	opprmulg 13627	. . . 4 |- ( ( R e. Ring /\ ( N (.g ` (oppr ` R ) ) Y ) e. B /\ X e. B ) -> ( ( N (.g ` (oppr ` R ) ) Y ) ( .r ` (oppr ` R ) ) X ) = ( X .X. ( N (.g ` (oppr ` R ) ) Y ) ) )
23	17, 20, 10, 22	syl3anc 1249	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( ( N (.g ` (oppr ` R ) ) Y ) ( .r ` (oppr ` R ) ) X ) = ( X .X. ( N (.g ` (oppr ` R ) ) Y ) ) )
24	6, 21, 1, 14	opprmulg 13627	. . . . 5 |- ( ( R e. Ring /\ Y e. B /\ X e. B ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .X. Y ) )
25	17, 5, 10, 24	syl3anc 1249	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .X. Y ) )
26	25	oveq2d 5938	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N (.g ` (oppr ` R ) ) ( Y ( .r ` (oppr ` R ) ) X ) ) = ( N (.g ` (oppr ` R ) ) ( X .X. Y ) ) )
27	16, 23, 26	3eqtr3d 2237	. 2 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( X .X. ( N (.g ` (oppr ` R ) ) Y ) ) = ( N (.g ` (oppr ` R ) ) ( X .X. Y ) ) )
28		mulgass3.m	. . . . . 6 |- .x. = (.g ` R )
29	28	a1i 9	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> .x. = (.g ` R ) )
30		eqidd 2197	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> (.g ` (oppr ` R ) ) = (.g ` (oppr ` R ) ) )
31	6	a1i 9	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> B = ( Base ` R ) )
32		ssidd 3204	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> B C_ B )
33		eqid 2196	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
34	6, 33	ringacl 13586	. . . . . . 7 |- ( ( R e. Ring /\ x e. B /\ y e. B ) -> ( x ( +g ` R ) y ) e. B )
35	34	3expb 1206	. . . . . 6 |- ( ( R e. Ring /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` R ) y ) e. B )
36	35	adantlr 477	. . . . 5 |- ( ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` R ) y ) e. B )
37	1, 33	oppraddg 13632	. . . . . . 7 |- ( R e. Ring -> ( +g ` R ) = ( +g ` (oppr ` R ) ) )
38	37	oveqdr 5950	. . . . . 6 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` R ) ) y ) )
39	38	adantr 276	. . . . 5 |- ( ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` R ) ) y ) )
40	29, 30, 17, 3, 31, 8, 32, 36, 39	mulgpropdg 13294	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> .x. = (.g ` (oppr ` R ) ) )
41	40	oveqd 5939	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N .x. Y ) = ( N (.g ` (oppr ` R ) ) Y ) )
42	41	oveq2d 5938	. 2 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( X .X. ( N .x. Y ) ) = ( X .X. ( N (.g ` (oppr ` R ) ) Y ) ) )
43	40	oveqd 5939	. 2 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N .x. ( X .X. Y ) ) = ( N (.g ` (oppr ` R ) ) ( X .X. Y ) ) )
44	27, 42, 43	3eqtr4d 2239	1 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( X .X. ( N .x. Y ) ) = ( N .x. ( X .X. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   ZZcz 9326   Basecbs 12678   +g cplusg 12755   .rcmulr 12756  ._gcmg 13249   Ringcrg 13552  opp_rcoppr 13623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-tpos 6303  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-3 9050  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-seqfrec 10540  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-mulg 13250  df-mgp 13477  df-ur 13516  df-ring 13554  df-oppr 13624

This theorem is referenced by: (None)