Theorem ringsubdir 16107

Description: Ring multiplication distributes over subtraction.

Hypotheses

Ref	Expression
ringsubdi.1	|- G = (1st` R)
ringsubdi.2	|- H = (2nd` R)
ringsubdi.3	|- X = ran G
ringsubdi.4	|- D = ( /g ` G)

Assertion

Ref	Expression
ringsubdir	|- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)HC) = ((AHC)D(BHC)))

Proof of Theorem ringsubdir

Step	Hyp	Ref	Expression
1		idd 75	. . . . . 6 |- (R e. Ring -> (A e. X -> A e. X))
2		ringsubdi.1	. . . . . . . 8 |- G = (1st` R)
3		ringsubdi.3	. . . . . . . 8 |- X = ran G
4		eqid 1884	. . . . . . . 8 |- (inv` G) = (inv` G)
5	2, 3, 4	ringnegcl 16098	. . . . . . 7 |- ((R e. Ring /\ B e. X) -> ((inv` G)` B) e. X)
6	5	ex 402	. . . . . 6 |- (R e. Ring -> (B e. X -> ((inv` G)` B) e. X))
7		idd 75	. . . . . 6 |- (R e. Ring -> (C e. X -> C e. X))
8	1, 6, 7	3anim123d 1175	. . . . 5 |- (R e. Ring -> ((A e. X /\ B e. X /\ C e. X) -> (A e. X /\ ((inv` G)` B) e. X /\ C e. X)))
9	8	imp 377	. . . 4 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> (A e. X /\ ((inv` G)` B) e. X /\ C e. X))
10		ringsubdi.2	. . . . 5 |- H = (2nd` R)
11	2, 10, 3	ringdir 9472	. . . 4 |- ((R e. Ring /\ (A e. X /\ ((inv` G)` B) e. X /\ C e. X)) -> ((AG((inv` G)` B))HC) = ((AHC)G(((inv` G)` B)HC)))
12	9, 11	syldan 516	. . 3 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AG((inv` G)` B))HC) = ((AHC)G(((inv` G)` B)HC)))
13	2, 10, 3, 4	ringneglmul 16104	. . . . 5 |- ((R e. Ring /\ B e. X /\ C e. X) -> ((inv` G)` (BHC)) = (((inv` G)` B)HC))
14	13	3adant3r1 1077	. . . 4 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((inv` G)` (BHC)) = (((inv` G)` B)HC))
15	14	opreq2d 4898	. . 3 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AHC)G((inv` G)` (BHC))) = ((AHC)G(((inv` G)` B)HC)))
16	12, 15	eqtr4d 1928	. 2 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AG((inv` G)` B))HC) = ((AHC)G((inv` G)` (BHC))))
17		ringsubdi.4	. . . . 5 |- D = ( /g ` G)
18	2, 3, 4, 17	ringsub 16101	. . . 4 |- ((R e. Ring /\ A e. X /\ B e. X) -> (ADB) = (AG((inv` G)` B)))
19	18	3adant3r3 1079	. . 3 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) = (AG((inv` G)` B)))
20	19	opreq1d 4897	. 2 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)HC) = ((AG((inv` G)` B))HC))
21	2, 10, 3	ringcl 9468	. . . . 5 |- ((R e. Ring /\ A e. X /\ C e. X) -> (AHC) e. X)
22	21	3adant3r2 1078	. . . 4 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> (AHC) e. X)
23	2, 10, 3	ringcl 9468	. . . . 5 |- ((R e. Ring /\ B e. X /\ C e. X) -> (BHC) e. X)
24	23	3adant3r1 1077	. . . 4 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> (BHC) e. X)
25	22, 24	jca 310	. . 3 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AHC) e. X /\ (BHC) e. X))
26	2, 3, 4, 17	ringsub 16101	. . . 4 |- ((R e. Ring /\ (AHC) e. X /\ (BHC) e. X) -> ((AHC)D(BHC)) = ((AHC)G((inv` G)` (BHC))))
27	26	3expb 1068	. . 3 |- ((R e. Ring /\ ((AHC) e. X /\ (BHC) e. X)) -> ((AHC)D(BHC)) = ((AHC)G((inv` G)` (BHC))))
28	25, 27	syldan 516	. 2 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AHC)D(BHC)) = ((AHC)G((inv` G)` (BHC))))
29	16, 20, 28	3eqtr4d 1937	1 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)HC) = ((AHC)D(BHC)))

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  invcgn 9313   /g cgs 9314  Ringcring 9463

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-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  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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