This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lduallkr3

Description: The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015)

Ref Expression
Hypotheses lduallkr3.h ⊢ 𝐻 = ( LSHyp ‘ 𝑊 )
lduallkr3.f ⊢ 𝐹 = ( LFnl ‘ 𝑊 )
lduallkr3.k ⊢ 𝐾 = ( LKer ‘ 𝑊 )
lduallkr3.d ⊢ 𝐷 = ( LDual ‘ 𝑊 )
lduallkr3.o ⊢ 0 = ( 0g ‘ 𝐷 )
lduallkr3.w ⊢ ( 𝜑 → 𝑊 ∈ LVec )
lduallkr3.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion lduallkr3 ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ↔ 𝐺 ≠ 0 ) )

Proof

Step Hyp Ref Expression
1 lduallkr3.h ⊢ 𝐻 = ( LSHyp ‘ 𝑊 )
2 lduallkr3.f ⊢ 𝐹 = ( LFnl ‘ 𝑊 )
3 lduallkr3.k ⊢ 𝐾 = ( LKer ‘ 𝑊 )
4 lduallkr3.d ⊢ 𝐷 = ( LDual ‘ 𝑊 )
5 lduallkr3.o ⊢ 0 = ( 0g ‘ 𝐷 )
6 lduallkr3.w ⊢ ( 𝜑 → 𝑊 ∈ LVec )
7 lduallkr3.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
8 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
9 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
10 eqid ⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
11 8 9 10 1 2 3 6 7 lkrshp3 ⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ↔ 𝐺 ≠ ( ( Base ‘ 𝑊 ) × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) )
12 lveclmod ⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod )
13 6 12 syl ⊢ ( 𝜑 → 𝑊 ∈ LMod )
14 8 9 10 4 5 13 ldual0v ⊢ ( 𝜑 → 0 = ( ( Base ‘ 𝑊 ) × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) )
15 14 neeq2d ⊢ ( 𝜑 → ( 𝐺 ≠ 0 ↔ 𝐺 ≠ ( ( Base ‘ 𝑊 ) × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) )
16 11 15 bitr4d ⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ∈ 𝐻 ↔ 𝐺 ≠ 0 ) )