Theorem ringnegrmul 16105

Description: Negation of a product in a ring.

Hypotheses

Ref	Expression
ringnegmul.1	|- G = (1st` R)
ringnegmul.2	|- H = (2nd` R)
ringnegmul.3	|- X = ran G
ringnegmul.4	|- N = (inv` G)

Assertion

Ref	Expression
ringnegrmul	|- ((R e. Ring /\ A e. X /\ B e. X) -> (N` (AHB)) = (AH(N` B)))

Proof of Theorem ringnegrmul

Step	Hyp	Ref	Expression
1		ringnegmul.3	. . . . . . 7 |- X = ran G
2		ringnegmul.1	. . . . . . . 8 |- G = (1st` R)
3	2	rneqi 4187	. . . . . . 7 |- ran G = ran (1st` R)
4	1, 3	eqtri 1908	. . . . . 6 |- X = ran (1st` R)
5		ringnegmul.2	. . . . . 6 |- H = (2nd` R)
6		eqid 1884	. . . . . 6 |- (Id` H) = (Id` H)
7	4, 5, 6	ring1cl 10415	. . . . 5 |- (R e. Ring -> (Id` H) e. X)
8		ringnegmul.4	. . . . . 6 |- N = (inv` G)
9	2, 1, 8	ringnegcl 16098	. . . . 5 |- ((R e. Ring /\ (Id` H) e. X) -> (N` (Id` H)) e. X)
10	7, 9	mpdan 768	. . . 4 |- (R e. Ring -> (N` (Id` H)) e. X)
11	2, 5, 1	ringass 9473	. . . . . . 7 |- ((R e. Ring /\ (A e. X /\ B e. X /\ (N` (Id` H)) e. X)) -> ((AHB)H(N` (Id` H))) = (AH(BH(N` (Id` H)))))
12	11	3exp2 1086	. . . . . 6 |- (R e. Ring -> (A e. X -> (B e. X -> ((N` (Id` H)) e. X -> ((AHB)H(N` (Id` H))) = (AH(BH(N` (Id` H))))))))
13	12	com24 41	. . . . 5 |- (R e. Ring -> ((N` (Id` H)) e. X -> (B e. X -> (A e. X -> ((AHB)H(N` (Id` H))) = (AH(BH(N` (Id` H))))))))
14	13	com34 40	. . . 4 |- (R e. Ring -> ((N` (Id` H)) e. X -> (A e. X -> (B e. X -> ((AHB)H(N` (Id` H))) = (AH(BH(N` (Id` H))))))))
15	10, 14	mpd 29	. . 3 |- (R e. Ring -> (A e. X -> (B e. X -> ((AHB)H(N` (Id` H))) = (AH(BH(N` (Id` H)))))))
16	15	3imp 1061	. 2 |- ((R e. Ring /\ A e. X /\ B e. X) -> ((AHB)H(N` (Id` H))) = (AH(BH(N` (Id` H)))))
17	2, 5, 1	ringcl 9468	. . . . 5 |- ((R e. Ring /\ A e. X /\ B e. X) -> (AHB) e. X)
18	17	3expb 1068	. . . 4 |- ((R e. Ring /\ (A e. X /\ B e. X)) -> (AHB) e. X)
19	2, 5, 1, 8, 6	ringnegmn1r 16103	. . . 4 |- ((R e. Ring /\ (AHB) e. X) -> (N` (AHB)) = ((AHB)H(N` (Id` H))))
20	18, 19	syldan 516	. . 3 |- ((R e. Ring /\ (A e. X /\ B e. X)) -> (N` (AHB)) = ((AHB)H(N` (Id` H))))
21	20	3impb 1063	. 2 |- ((R e. Ring /\ A e. X /\ B e. X) -> (N` (AHB)) = ((AHB)H(N` (Id` H))))
22	2, 5, 1, 8, 6	ringnegmn1r 16103	. . . 4 |- ((R e. Ring /\ B e. X) -> (N` B) = (BH(N` (Id` H))))
23	22	3adant2 895	. . 3 |- ((R e. Ring /\ A e. X /\ B e. X) -> (N` B) = (BH(N` (Id` H))))
24	23	opreq2d 4898	. 2 |- ((R e. Ring /\ A e. X /\ B e. X) -> (AH(N` B)) = (AH(BH(N` (Id` H)))))
25	16, 21, 24	3eqtr4d 1937	1 |- ((R e. Ring /\ A e. X /\ B e. X) -> (N` (AHB)) = (AH(N` B)))

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

This theorem is referenced by:  ringsubdi 16106

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  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385

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