Theorem lvecpropd

Description: If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015)

	Ref	Expression
Hypotheses	lvecpropd.1	|- ( ph -> B = ( Base ` K ) )
	lvecpropd.2	|- ( ph -> B = ( Base ` L ) )
	lvecpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
	lvecpropd.4	|- ( ph -> F = ( Scalar ` K ) )
	lvecpropd.5	|- ( ph -> F = ( Scalar ` L ) )
	lvecpropd.6	|- P = ( Base ` F )
	lvecpropd.7	|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )
Assertion	lvecpropd	|- ( ph -> ( K e. LVec <-> L e. LVec ) )

Proof

Step	Hyp	Ref	Expression
1		lvecpropd.1	|- ( ph -> B = ( Base ` K ) )
2		lvecpropd.2	|- ( ph -> B = ( Base ` L ) )
3		lvecpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
4		lvecpropd.4	|- ( ph -> F = ( Scalar ` K ) )
5		lvecpropd.5	|- ( ph -> F = ( Scalar ` L ) )
6		lvecpropd.6	|- P = ( Base ` F )
7		lvecpropd.7	|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )
8	1 2 3 4 5 6 7	lmodpropd	|- ( ph -> ( K e. LMod <-> L e. LMod ) )
9	4 5	eqtr3d	|- ( ph -> ( Scalar ` K ) = ( Scalar ` L ) )
10	9	eleq1d	|- ( ph -> ( ( Scalar ` K ) e. DivRing <-> ( Scalar ` L ) e. DivRing ) )
11	8 10	anbi12d	|- ( ph -> ( ( K e. LMod /\ ( Scalar ` K ) e. DivRing ) <-> ( L e. LMod /\ ( Scalar ` L ) e. DivRing ) ) )
12		eqid	|- ( Scalar ` K ) = ( Scalar ` K )
13	12	islvec	|- ( K e. LVec <-> ( K e. LMod /\ ( Scalar ` K ) e. DivRing ) )
14		eqid	|- ( Scalar ` L ) = ( Scalar ` L )
15	14	islvec	|- ( L e. LVec <-> ( L e. LMod /\ ( Scalar ` L ) e. DivRing ) )
16	11 13 15	3bitr4g	|- ( ph -> ( K e. LVec <-> L e. LVec ) )