Theorem multinv 14771

Description: Multiplication by an additive inverse.

Hypotheses

Ref	Expression
multinv.1	|- X = ran G
multinv.2	|- G = (1st` R)
multinv.3	|- H = (2nd` R)

Assertion

Ref	Expression
multinv	|- ((R e. Ring /\ A e. X /\ B e. X) -> (((inv` G)` A)HB) = ((inv` G)` (AHB)))

Proof of Theorem multinv

Step	Hyp	Ref	Expression
1		simp1 876	. . . . 5 |- ((R e. Ring /\ A e. X /\ B e. X) -> R e. Ring)
2		simp2 877	. . . . 5 |- ((R e. Ring /\ A e. X /\ B e. X) -> A e. X)
3		multinv.2	. . . . . . . . 9 |- G = (1st` R)
4	3	ringgrp 9476	. . . . . . . 8 |- (R e. Ring -> G e. Grp)
5		multinv.1	. . . . . . . . . 10 |- X = ran G
6		eqid 1884	. . . . . . . . . 10 |- (inv` G) = (inv` G)
7	5, 6	grpinvcl 9352	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> ((inv` G)` A) e. X)
8	7	ex 402	. . . . . . . 8 |- (G e. Grp -> (A e. X -> ((inv` G)` A) e. X))
9	4, 8	syl 12	. . . . . . 7 |- (R e. Ring -> (A e. X -> ((inv` G)` A) e. X))
10	9	a1dd 53	. . . . . 6 |- (R e. Ring -> (A e. X -> (B e. X -> ((inv` G)` A) e. X)))
11	10	3imp 1061	. . . . 5 |- ((R e. Ring /\ A e. X /\ B e. X) -> ((inv` G)` A) e. X)
12		simp3 878	. . . . 5 |- ((R e. Ring /\ A e. X /\ B e. X) -> B e. X)
13		multinv.3	. . . . . . 7 |- H = (2nd` R)
14	3, 13, 5	ringdir 9472	. . . . . 6 |- ((R e. Ring /\ (A e. X /\ ((inv` G)` A) e. X /\ B e. X)) -> ((AG((inv` G)` A))HB) = ((AHB)G(((inv` G)` A)HB)))
15	14	eqcomd 1889	. . . . 5 |- ((R e. Ring /\ (A e. X /\ ((inv` G)` A) e. X /\ B e. X)) -> ((AHB)G(((inv` G)` A)HB)) = ((AG((inv` G)` A))HB))
16	1, 2, 11, 12, 15	syl13anc 1102	. . . 4 |- ((R e. Ring /\ A e. X /\ B e. X) -> ((AHB)G(((inv` G)` A)HB)) = ((AG((inv` G)` A))HB))
17	4	3ad2ant1 897	. . . . . 6 |- ((R e. Ring /\ A e. X /\ B e. X) -> G e. Grp)
18		eqid 1884	. . . . . . 7 |- (Id` G) = (Id` G)
19	5, 18, 6	grprinv 9355	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (AG((inv` G)` A)) = (Id` G))
20	17, 2, 19	syl11anc 524	. . . . 5 |- ((R e. Ring /\ A e. X /\ B e. X) -> (AG((inv` G)` A)) = (Id` G))
21	20	opreq1d 4897	. . . 4 |- ((R e. Ring /\ A e. X /\ B e. X) -> ((AG((inv` G)` A))HB) = ((Id` G)HB))
22	18, 5, 3, 13	ringlz 9487	. . . . 5 |- ((R e. Ring /\ B e. X) -> ((Id` G)HB) = (Id` G))
23	22	3adant2 895	. . . 4 |- ((R e. Ring /\ A e. X /\ B e. X) -> ((Id` G)HB) = (Id` G))
24	16, 21, 23	3eqtrd 1929	. . 3 |- ((R e. Ring /\ A e. X /\ B e. X) -> ((AHB)G(((inv` G)` A)HB)) = (Id` G))
25	3, 13, 5	ringcl 9468	. . . 4 |- ((R e. Ring /\ A e. X /\ B e. X) -> (AHB) e. X)
26	3, 13, 5	ringcl 9468	. . . . 5 |- ((R e. Ring /\ ((inv` G)` A) e. X /\ B e. X) -> (((inv` G)` A)HB) e. X)
27	26, 11	syld3an2 1144	. . . 4 |- ((R e. Ring /\ A e. X /\ B e. X) -> (((inv` G)` A)HB) e. X)
28	5, 18, 6	grpinvid1 9356	. . . 4 |- ((G e. Grp /\ (AHB) e. X /\ (((inv` G)` A)HB) e. X) -> (((inv` G)` (AHB)) = (((inv` G)` A)HB) <-> ((AHB)G(((inv` G)` A)HB)) = (Id` G)))
29	17, 25, 27, 28	syl111anc 1100	. . 3 |- ((R e. Ring /\ A e. X /\ B e. X) -> (((inv` G)` (AHB)) = (((inv` G)` A)HB) <-> ((AHB)G(((inv` G)` A)HB)) = (Id` G)))
30	24, 29	mpbird 213	. 2 |- ((R e. Ring /\ A e. X /\ B e. X) -> ((inv` G)` (AHB)) = (((inv` G)` A)HB))
31	30	eqcomd 1889	1 |- ((R e. Ring /\ A e. X /\ B e. X) -> (((inv` G)` A)HB) = ((inv` G)` (AHB)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  invcgn 9313  Ringcring 9463

This theorem is referenced by:  mult2inv 14773

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-ring 9464

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