Theorem fldpropd

Description: If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015)

	Ref	Expression
Hypotheses	drngpropd.1	|- ( ph -> B = ( Base ` K ) )
	drngpropd.2	|- ( ph -> B = ( Base ` L ) )
	drngpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
	drngpropd.4	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
Assertion	fldpropd	|- ( ph -> ( K e. Field <-> L e. Field ) )

Proof

Step	Hyp	Ref	Expression
1		drngpropd.1	|- ( ph -> B = ( Base ` K ) )
2		drngpropd.2	|- ( ph -> B = ( Base ` L ) )
3		drngpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
4		drngpropd.4	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
5	1 2 3 4	drngpropd	|- ( ph -> ( K e. DivRing <-> L e. DivRing ) )
6	1 2 3 4	crngpropd	|- ( ph -> ( K e. CRing <-> L e. CRing ) )
7	5 6	anbi12d	|- ( ph -> ( ( K e. DivRing /\ K e. CRing ) <-> ( L e. DivRing /\ L e. CRing ) ) )
8		isfld	|- ( K e. Field <-> ( K e. DivRing /\ K e. CRing ) )
9		isfld	|- ( L e. Field <-> ( L e. DivRing /\ L e. CRing ) )
10	7 8 9	3bitr4g	|- ( ph -> ( K e. Field <-> L e. Field ) )