Theorem mulgass3 13581

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 2193	. . . . . 6 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13575	. . . . 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 13571	. . . . . 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 2272	. . . 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 2272	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> X e. ( Base ` (oppr ` R ) ) )
12		eqid 2193	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
13		eqid 2193	. . . . 5 |- (.g ` (oppr ` R ) ) = (.g ` (oppr ` R ) )
14		eqid 2193	. . . . 5 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
15	12, 13, 14	mulgass2 13554	. . . 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 13501	. . . . . 6 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> (oppr ` R ) e. Grp )
19	12, 13, 18, 4, 9	mulgcld 13214	. . . . 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 2273	. . . 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 13567	. . . 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 13567	. . . . 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 5934	. . 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 2234	. 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 2194	. . . . 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 3200	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> B C_ B )
33		eqid 2193	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
34	6, 33	ringacl 13526	. . . . . . 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 13572	. . . . . . 7 |- ( R e. Ring -> ( +g ` R ) = ( +g ` (oppr ` R ) ) )
38	37	oveqdr 5946	. . . . . 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 13234	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> .x. = (.g ` (oppr ` R ) ) )
41	40	oveqd 5935	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N .x. Y ) = ( N (.g ` (oppr ` R ) ) Y ) )
42	41	oveq2d 5934	. 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 5935	. 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 2236	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 2164   ` cfv 5254  (class class class)co 5918   ZZcz 9317   Basecbs 12618   +g cplusg 12695   .rcmulr 12696  ._gcmg 13189   Ringcrg 13492  opp_rcoppr 13563

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-tpos 6298  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-3 9042  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-mulg 13190  df-mgp 13417  df-ur 13456  df-ring 13494  df-oppr 13564

This theorem is referenced by: (None)