Theorem mulgass3 13396

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 2189	. . . . . 6 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13390	. . . . 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 13386	. . . . . 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 2268	. . . 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 2268	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> X e. ( Base ` (oppr ` R ) ) )
12		eqid 2189	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
13		eqid 2189	. . . . 5 |- (.g ` (oppr ` R ) ) = (.g ` (oppr ` R ) )
14		eqid 2189	. . . . 5 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
15	12, 13, 14	mulgass2 13371	. . . 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 13320	. . . . . 6 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> (oppr ` R ) e. Grp )
19	12, 13, 18, 4, 9	mulgcld 13050	. . . . 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 2269	. . . 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 13382	. . . 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 13382	. . . . 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 5907	. . 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 2230	. 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 2190	. . . . 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 3191	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> B C_ B )
33		eqid 2189	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
34	6, 33	ringacl 13345	. . . . . . 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 13387	. . . . . . 7 |- ( R e. Ring -> ( +g ` R ) = ( +g ` (oppr ` R ) ) )
38	37	oveqdr 5919	. . . . . 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 13070	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> .x. = (.g ` (oppr ` R ) ) )
41	40	oveqd 5908	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N .x. Y ) = ( N (.g ` (oppr ` R ) ) Y ) )
42	41	oveq2d 5907	. 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 5908	. 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 2232	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 2160   ` cfv 5231  (class class class)co 5891   ZZcz 9271   Basecbs 12480   +g cplusg 12555   .rcmulr 12556  ._gcmg 13027   Ringcrg 13311  opp_rcoppr 13378

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-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-ltadd 7945

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 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-reu 2475  df-rmo 2476  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-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-tpos 6264  df-recs 6324  df-frec 6410  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-inn 8938  df-2 8996  df-3 8997  df-n0 9195  df-z 9272  df-uz 9547  df-fz 10027  df-seqfrec 10464  df-ndx 12483  df-slot 12484  df-base 12486  df-sets 12487  df-plusg 12568  df-mulr 12569  df-0g 12729  df-mgm 12798  df-sgrp 12831  df-mnd 12844  df-grp 12914  df-minusg 12915  df-mulg 13028  df-mgp 13236  df-ur 13275  df-ring 13313  df-oppr 13379

This theorem is referenced by: (None)