Theorem mulgass3 14364

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 2238	. . . . . 6 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 14357	. . . . 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 1034	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> N e. ZZ )
5		simpr3 1036	. . . . 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 14353	. . . . . 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 2317	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> Y e. ( Base ` (oppr ` R ) ) )
10		simpr2 1035	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> X e. B )
11	10, 8	eleqtrd 2317	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> X e. ( Base ` (oppr ` R ) ) )
12		eqid 2238	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
13		eqid 2238	. . . . 5 |- (.g ` (oppr ` R ) ) = (.g ` (oppr ` R ) )
14		eqid 2238	. . . . 5 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
15	12, 13, 14	mulgass2 14336	. . . 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 1280	. . 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 14283	. . . . . 6 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> (oppr ` R ) e. Grp )
19	12, 13, 18, 4, 9	mulgcld 13924	. . . . 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 2318	. . . 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 14349	. . . 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 1278	. . 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 14349	. . . . 5 |- ( ( R e. Ring /\ Y e. B /\ X e. B ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .X. Y ) )
25	17, 5, 10, 24	syl3anc 1278	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( Y ( .r ` (oppr ` R ) ) X ) = ( X .X. Y ) )
26	25	oveq2d 6091	. . 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 2279	. 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 2239	. . . . 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 3269	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> B C_ B )
33		eqid 2238	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
34	6, 33	ringacl 14308	. . . . . . 7 |- ( ( R e. Ring /\ x e. B /\ y e. B ) -> ( x ( +g ` R ) y ) e. B )
35	34	3expb 1235	. . . . . 6 |- ( ( R e. Ring /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` R ) y ) e. B )
36	35	adantlr 481	. . . . 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 14354	. . . . . . 7 |- ( R e. Ring -> ( +g ` R ) = ( +g ` (oppr ` R ) ) )
38	37	oveqdr 6103	. . . . . 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 13944	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> .x. = (.g ` (oppr ` R ) ) )
41	40	oveqd 6092	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N .x. Y ) = ( N (.g ` (oppr ` R ) ) Y ) )
42	41	oveq2d 6091	. 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 6092	. 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 2281	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 1009   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   ZZcz 9623   Basecbs 13330   +g cplusg 13408   .rcmulr 13409  ._gcmg 13899   Ringcrg 14274  opp_rcoppr 14345

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-tpos 6506  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-mulg 13900  df-mgp 14195  df-ur 14238  df-ring 14276  df-oppr 14346

This theorem is referenced by: (None)