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Metamath Proof Explorer

Theorem lkr0f2

Description: The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015)

Ref Expression
Hypotheses lkr0f2.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lkr0f2.f ⊢ 𝐹 = ( LFnl ‘ 𝑊 )
lkr0f2.k ⊢ 𝐾 = ( LKer ‘ 𝑊 )
lkr0f2.d ⊢ 𝐷 = ( LDual ‘ 𝑊 )
lkr0f2.o ⊢ 0 = ( 0g ‘ 𝐷 )
lkr0f2.w ⊢ ( 𝜑 → 𝑊 ∈ LMod )
lkr0f2.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion lkr0f2 ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) = 𝑉 ↔ 𝐺 = 0 ) )

Proof

Step Hyp Ref Expression
1 lkr0f2.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lkr0f2.f ⊢ 𝐹 = ( LFnl ‘ 𝑊 )
3 lkr0f2.k ⊢ 𝐾 = ( LKer ‘ 𝑊 )
4 lkr0f2.d ⊢ 𝐷 = ( LDual ‘ 𝑊 )
5 lkr0f2.o ⊢ 0 = ( 0g ‘ 𝐷 )
6 lkr0f2.w ⊢ ( 𝜑 → 𝑊 ∈ LMod )
7 lkr0f2.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
8 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
9 eqid ⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
10 8 9 1 2 3 lkr0f ⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( ( 𝐾 ‘ 𝐺 ) = 𝑉 ↔ 𝐺 = ( 𝑉 × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) )
11 6 7 10 syl2anc ⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) = 𝑉 ↔ 𝐺 = ( 𝑉 × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) )
12 1 8 9 4 5 6 ldual0v ⊢ ( 𝜑 → 0 = ( 𝑉 × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) )
13 12 eqeq2d ⊢ ( 𝜑 → ( 𝐺 = 0 ↔ 𝐺 = ( 𝑉 × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) )
14 11 13 bitr4d ⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) = 𝑉 ↔ 𝐺 = 0 ) )