Theorem ring2 8149

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 8145	. . 3 |- ((R e. Ring /\ A e. X) -> E.x e. X ((AHx) = A /\ (xHA) = A))
5		pm3.27 323	. . . 4 |- (((AHx) = A /\ (xHA) = A) -> (xHA) = A)
6	5	r19.22si 1734	. . 3 |- (E.x e. X ((AHx) = A /\ (xHA) = A) -> E.x e. X (xHA) = A)
7		opreq12 3970	. . . . 5 |- (((xHA) = A /\ (xHA) = A) -> ((xHA)G(xHA)) = (AGA))
8	7	anidms 434	. . . 4 |- ((xHA) = A -> ((xHA)G(xHA)) = (AGA))
9	8	r19.22si 1734	. . 3 |- (E.x e. X (xHA) = A -> E.x e. X ((xHA)G(xHA)) = (AGA))
10	4, 6, 9	3syl 20	. 2 |- ((R e. Ring /\ A e. X) -> E.x e. X ((xHA)G(xHA)) = (AGA))
11		eqtrt 1492	. . . . . . 7 |- ((((xGx)HA) = ((xHA)G(xHA)) /\ ((xHA)G(xHA)) = (AGA)) -> ((xGx)HA) = (AGA))
12	11	eqcomd 1480	. . . . . 6 |- ((((xGx)HA) = ((xHA)G(xHA)) /\ ((xHA)G(xHA)) = (AGA)) -> (AGA) = ((xGx)HA))
13	1, 2, 3	ringdir 8147	. . . . . . . . . . 11 |- ((R e. Ring /\ (x e. X /\ x e. X /\ A e. X)) -> ((xGx)HA) = ((xHA)G(xHA)))
14	13	expcom 374	. . . . . . . . . 10 |- ((x e. X /\ x e. X /\ A e. X) -> (R e. Ring -> ((xGx)HA) = ((xHA)G(xHA))))
15	14	3expia 835	. . . . . . . . 9 |- ((x e. X /\ x e. X) -> (A e. X -> (R e. Ring -> ((xGx)HA) = ((xHA)G(xHA)))))
16	15	anidms 434	. . . . . . . 8 |- (x e. X -> (A e. X -> (R e. Ring -> ((xGx)HA) = ((xHA)G(xHA)))))
17	16	3imp 827	. . . . . . 7 |- ((x e. X /\ A e. X /\ R e. Ring) -> ((xGx)HA) = ((xHA)G(xHA)))
18	17	3com13 838	. . . . . 6 |- ((R e. Ring /\ A e. X /\ x e. X) -> ((xGx)HA) = ((xHA)G(xHA)))
19	12, 18	sylan 448	. . . . 5 |- (((R e. Ring /\ A e. X /\ x e. X) /\ ((xHA)G(xHA)) = (AGA)) -> (AGA) = ((xGx)HA))
20	19	ex 373	. . . 4 |- ((R e. Ring /\ A e. X /\ x e. X) -> (((xHA)G(xHA)) = (AGA) -> (AGA) = ((xGx)HA)))
21	20	3expa 833	. . 3 |- (((R e. Ring /\ A e. X) /\ x e. X) -> (((xHA)G(xHA)) = (AGA) -> (AGA) = ((xGx)HA)))
22	21	r19.22dva 1739	. 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 26	1 |- ((R e. Ring /\ A e. X) -> E.x e. X (AGA) = ((xGx)HA))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wrex 1646  ran crn 3171  ` cfv 3182  (class class class)co 3963  1stc1st 4077  2ndc2nd 4078  Ringcring 8139

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-1st 4079  df-2nd 4080  df-ring 8140

Copyright terms: Public domain