Theorem mulgass3 13818

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 2204	. . . . . 6 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13812	. . . . 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 13808	. . . . . 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 2283	. . . 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 2283	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> X e. ( Base ` (oppr ` R ) ) )
12		eqid 2204	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
13		eqid 2204	. . . . 5 |- (.g ` (oppr ` R ) ) = (.g ` (oppr ` R ) )
14		eqid 2204	. . . . 5 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
15	12, 13, 14	mulgass2 13791	. . . 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 13738	. . . . . 6 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> (oppr ` R ) e. Grp )
19	12, 13, 18, 4, 9	mulgcld 13451	. . . . 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 2284	. . . 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 13804	. . . 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 13804	. . . . 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 5959	. . 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 2245	. 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 2205	. . . . 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 3213	. . . . 5 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> B C_ B )
33		eqid 2204	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
34	6, 33	ringacl 13763	. . . . . . 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 13809	. . . . . . 7 |- ( R e. Ring -> ( +g ` R ) = ( +g ` (oppr ` R ) ) )
38	37	oveqdr 5971	. . . . . 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 13471	. . . 4 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> .x. = (.g ` (oppr ` R ) ) )
41	40	oveqd 5960	. . 3 |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( N .x. Y ) = ( N (.g ` (oppr ` R ) ) Y ) )
42	41	oveq2d 5959	. 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 5960	. 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 2247	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 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   ZZcz 9371   Basecbs 12803   +g cplusg 12880   .rcmulr 12881  ._gcmg 13426   Ringcrg 13729  opp_rcoppr 13800

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-tpos 6330  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-2 9094  df-3 9095  df-n0 9295  df-z 9372  df-uz 9648  df-fz 10130  df-seqfrec 10591  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-plusg 12893  df-mulr 12894  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-mulg 13427  df-mgp 13654  df-ur 13693  df-ring 13731  df-oppr 13801

This theorem is referenced by: (None)