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Metamath Proof Explorer

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 ) )