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Metamath Proof Explorer

Theorem rngoid

Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by Mario Carneiro, 22-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses ringi.1 
|- G = ( 1st ` R )
ringi.2 
|- H = ( 2nd ` R )
ringi.3 
|- X = ran G
Assertion rngoid 
|- ( ( R e. RingOps /\ A e. X ) -> E. u e. X ( ( u H A ) = A /\ ( A H u ) = A ) )

Proof

Step Hyp Ref Expression
1 ringi.1 
 |-  G = ( 1st ` R )
2 ringi.2 
 |-  H = ( 2nd ` R )
3 ringi.3 
 |-  X = ran G
4 1 2 3 rngoi 
 |-  ( R e. RingOps -> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. u e. X A. x e. X A. y e. X ( ( ( u H x ) H y ) = ( u H ( x H y ) ) /\ ( u H ( x G y ) ) = ( ( u H x ) G ( u H y ) ) /\ ( ( u G x ) H y ) = ( ( u H y ) G ( x H y ) ) ) /\ E. u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) ) ) )
5 4 simprrd 
 |-  ( R e. RingOps -> E. u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) )
6 r19.12 
 |-  ( E. u e. X A. x e. X ( ( u H x ) = x /\ ( x H u ) = x ) -> A. x e. X E. u e. X ( ( u H x ) = x /\ ( x H u ) = x ) )
7 5 6 syl 
 |-  ( R e. RingOps -> A. x e. X E. u e. X ( ( u H x ) = x /\ ( x H u ) = x ) )
8 oveq2 
 |-  ( x = A -> ( u H x ) = ( u H A ) )
9 id 
 |-  ( x = A -> x = A )
10 8 9 eqeq12d 
 |-  ( x = A -> ( ( u H x ) = x <-> ( u H A ) = A ) )
11 oveq1 
 |-  ( x = A -> ( x H u ) = ( A H u ) )
12 11 9 eqeq12d 
 |-  ( x = A -> ( ( x H u ) = x <-> ( A H u ) = A ) )
13 10 12 anbi12d 
 |-  ( x = A -> ( ( ( u H x ) = x /\ ( x H u ) = x ) <-> ( ( u H A ) = A /\ ( A H u ) = A ) ) )
14 13 rexbidv 
 |-  ( x = A -> ( E. u e. X ( ( u H x ) = x /\ ( x H u ) = x ) <-> E. u e. X ( ( u H A ) = A /\ ( A H u ) = A ) ) )
15 14 rspccva 
 |-  ( ( A. x e. X E. u e. X ( ( u H x ) = x /\ ( x H u ) = x ) /\ A e. X ) -> E. u e. X ( ( u H A ) = A /\ ( A H u ) = A ) )
16 7 15 sylan 
 |-  ( ( R e. RingOps /\ A e. X ) -> E. u e. X ( ( u H A ) = A /\ ( A H u ) = A ) )