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

Metamath Proof Explorer

Theorem ldualkrsc

Description: The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014)

Ref Expression
Hypotheses ldualkrsc.r ⊢ 𝑅 = ( Scalar ‘ 𝑊 )
ldualkrsc.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
ldualkrsc.o ⊢ 0 = ( 0g ‘ 𝑅 )
ldualkrsc.f ⊢ 𝐹 = ( LFnl ‘ 𝑊 )
ldualkrsc.l ⊢ 𝐿 = ( LKer ‘ 𝑊 )
ldualkrsc.d ⊢ 𝐷 = ( LDual ‘ 𝑊 )
ldualkrsc.s ⊢ · = ( ·𝑠 ‘ 𝐷 )
ldualkrsc.w ⊢ ( 𝜑 → 𝑊 ∈ LVec )
ldualkrsc.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
ldualkrsc.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
ldualkrsc.e ⊢ ( 𝜑 → 𝑋 ≠ 0 )
Assertion ldualkrsc ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 ldualkrsc.r ⊢ 𝑅 = ( Scalar ‘ 𝑊 )
2 ldualkrsc.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
3 ldualkrsc.o ⊢ 0 = ( 0g ‘ 𝑅 )
4 ldualkrsc.f ⊢ 𝐹 = ( LFnl ‘ 𝑊 )
5 ldualkrsc.l ⊢ 𝐿 = ( LKer ‘ 𝑊 )
6 ldualkrsc.d ⊢ 𝐷 = ( LDual ‘ 𝑊 )
7 ldualkrsc.s ⊢ · = ( ·𝑠 ‘ 𝐷 )
8 ldualkrsc.w ⊢ ( 𝜑 → 𝑊 ∈ LVec )
9 ldualkrsc.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
10 ldualkrsc.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
11 ldualkrsc.e ⊢ ( 𝜑 → 𝑋 ≠ 0 )
12 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
13 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
14 4 12 1 2 13 6 7 8 10 9 ldualvs ⊢ ( 𝜑 → ( 𝑋 · 𝐺 ) = ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) )
15 14 fveq2d ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) )
16 12 1 2 13 4 5 8 9 10 3 11 lkrsc ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) = ( 𝐿 ‘ 𝐺 ) )
17 15 16 eqtrd ⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ 𝐺 ) )