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Metamath Proof Explorer

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 ) )