Theorem ring2 9474

Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Hypotheses

Ref	Expression
ring2.1	|- G = (1st` R)
ring2.2	|- H = (2nd` R)
ring2.3	|- X = ran G

Assertion

Ref	Expression
ring2	|- ((R e. Ring /\ A e. X) -> E.x e. X (AGA) = ((xGx)HA))

Distinct variable groups:   x,A   x,G   x,H   x,R   x,X

Proof of Theorem ring2

Step	Hyp	Ref	Expression
1		ring2.1	. . . 4 |- G = (1st` R)
2		ring2.2	. . . 4 |- H = (2nd` R)
3		ring2.3	. . . 4 |- X = ran G
4	1, 2, 3	ringid 9469	. . 3 |- ((R e. Ring /\ A e. X) -> E.x e. X ((AHx) = A /\ (xHA) = A))
5		simpr 350	. . . 4 |- (((AHx) = A /\ (xHA) = A) -> (xHA) = A)
6	5	reximi 2198	. . 3 |- (E.x e. X ((AHx) = A /\ (xHA) = A) -> E.x e. X (xHA) = A)
7		opreq12 4891	. . . . 5 |- (((xHA) = A /\ (xHA) = A) -> ((xHA)G(xHA)) = (AGA))
8	7	anidms 480	. . . 4 |- ((xHA) = A -> ((xHA)G(xHA)) = (AGA))
9	8	reximi 2198	. . 3 |- (E.x e. X (xHA) = A -> E.x e. X ((xHA)G(xHA)) = (AGA))
10	4, 6, 9	3syl 24	. 2 |- ((R e. Ring /\ A e. X) -> E.x e. X ((xHA)G(xHA)) = (AGA))
11		eqtr 1904	. . . . . . 7 |- ((((xGx)HA) = ((xHA)G(xHA)) /\ ((xHA)G(xHA)) = (AGA)) -> ((xGx)HA) = (AGA))
12	11	eqcomd 1889	. . . . . 6 |- ((((xGx)HA) = ((xHA)G(xHA)) /\ ((xHA)G(xHA)) = (AGA)) -> (AGA) = ((xGx)HA))
13	1, 2, 3	ringdir 9472	. . . . . . . . . . 11 |- ((R e. Ring /\ (x e. X /\ x e. X /\ A e. X)) -> ((xGx)HA) = ((xHA)G(xHA)))
14	13	expcom 403	. . . . . . . . . 10 |- ((x e. X /\ x e. X /\ A e. X) -> (R e. Ring -> ((xGx)HA) = ((xHA)G(xHA))))
15	14	3expia 1069	. . . . . . . . 9 |- ((x e. X /\ x e. X) -> (A e. X -> (R e. Ring -> ((xGx)HA) = ((xHA)G(xHA)))))
16	15	anidms 480	. . . . . . . 8 |- (x e. X -> (A e. X -> (R e. Ring -> ((xGx)HA) = ((xHA)G(xHA)))))
17	16	3imp 1061	. . . . . . 7 |- ((x e. X /\ A e. X /\ R e. Ring) -> ((xGx)HA) = ((xHA)G(xHA)))
18	17	3com13 1073	. . . . . 6 |- ((R e. Ring /\ A e. X /\ x e. X) -> ((xGx)HA) = ((xHA)G(xHA)))
19	12, 18	sylan 497	. . . . 5 |- (((R e. Ring /\ A e. X /\ x e. X) /\ ((xHA)G(xHA)) = (AGA)) -> (AGA) = ((xGx)HA))
20	19	ex 402	. . . 4 |- ((R e. Ring /\ A e. X /\ x e. X) -> (((xHA)G(xHA)) = (AGA) -> (AGA) = ((xGx)HA)))
21	20	3expa 1067	. . 3 |- (((R e. Ring /\ A e. X) /\ x e. X) -> (((xHA)G(xHA)) = (AGA) -> (AGA) = ((xGx)HA)))
22	21	reximdva 2203	. 2 |- ((R e. Ring /\ A e. X) -> (E.x e. X ((xHA)G(xHA)) = (AGA) -> E.x e. X (AGA) = ((xGx)HA)))
23	10, 22	mpd 29	1 |- ((R e. Ring /\ A e. X) -> E.x e. X (AGA) = ((xGx)HA))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  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-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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464

Copyright terms: Public domain