Theorem rngo2 33706

Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
rngo2	|- ( ( R e. RingOps /\ A e. X ) -> E. x e. X ( A G A ) = ( ( x G x ) H A ) )

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

Proof of Theorem rngo2

Step	Hyp	Ref	Expression
1		ringi.1	. . 3 |- G = ( 1st ` R )
2		ringi.2	. . 3 |- H = ( 2nd ` R )
3		ringi.3	. . 3 |- X = ran G
4	1, 2, 3	rngoid 33701	. 2 |- ( ( R e. RingOps /\ A e. X ) -> E. x e. X ( ( x H A ) = A /\ ( A H x ) = A ) )
5		oveq12 6659	. . . . . . 7 |- ( ( ( x H A ) = A /\ ( x H A ) = A ) -> ( ( x H A ) G ( x H A ) ) = ( A G A ) )
6	5	anidms 677	. . . . . 6 |- ( ( x H A ) = A -> ( ( x H A ) G ( x H A ) ) = ( A G A ) )
7	6	eqcomd 2628	. . . . 5 |- ( ( x H A ) = A -> ( A G A ) = ( ( x H A ) G ( x H A ) ) )
8		simpll 790	. . . . . . 7 |- ( ( ( R e. RingOps /\ A e. X ) /\ x e. X ) -> R e. RingOps )
9		simpr 477	. . . . . . 7 |- ( ( ( R e. RingOps /\ A e. X ) /\ x e. X ) -> x e. X )
10		simplr 792	. . . . . . 7 |- ( ( ( R e. RingOps /\ A e. X ) /\ x e. X ) -> A e. X )
11	1, 2, 3	rngodir 33704	. . . . . . 7 |- ( ( R e. RingOps /\ ( x e. X /\ x e. X /\ A e. X ) ) -> ( ( x G x ) H A ) = ( ( x H A ) G ( x H A ) ) )
12	8, 9, 9, 10, 11	syl13anc 1328	. . . . . 6 |- ( ( ( R e. RingOps /\ A e. X ) /\ x e. X ) -> ( ( x G x ) H A ) = ( ( x H A ) G ( x H A ) ) )
13	12	eqeq2d 2632	. . . . 5 |- ( ( ( R e. RingOps /\ A e. X ) /\ x e. X ) -> ( ( A G A ) = ( ( x G x ) H A ) <-> ( A G A ) = ( ( x H A ) G ( x H A ) ) ) )
14	7, 13	syl5ibr 236	. . . 4 |- ( ( ( R e. RingOps /\ A e. X ) /\ x e. X ) -> ( ( x H A ) = A -> ( A G A ) = ( ( x G x ) H A ) ) )
15	14	adantrd 484	. . 3 |- ( ( ( R e. RingOps /\ A e. X ) /\ x e. X ) -> ( ( ( x H A ) = A /\ ( A H x ) = A ) -> ( A G A ) = ( ( x G x ) H A ) ) )
16	15	reximdva 3017	. 2 |- ( ( R e. RingOps /\ A e. X ) -> ( E. x e. X ( ( x H A ) = A /\ ( A H x ) = A ) -> E. x e. X ( A G A ) = ( ( x G x ) H A ) ) )
17	4, 16	mpd 15	1 |- ( ( R e. RingOps /\ A e. X ) -> E. x e. X ( A G A ) = ( ( x G x ) H A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694

This theorem is referenced by: (None)