Theorem dochkrshp2

Description: Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015)

	Ref	Expression
Hypotheses	dochkrshp2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dochkrshp2.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	dochkrshp2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dochkrshp2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	dochkrshp2.y	⊢ 𝑌 = ( LSHyp ‘ 𝑈 )
	dochkrshp2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	dochkrshp2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	dochkrshp2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dochkrshp2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion	dochkrshp2	⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dochkrshp2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dochkrshp2.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		dochkrshp2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		dochkrshp2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		dochkrshp2.y	⊢ 𝑌 = ( LSHyp ‘ 𝑈 )
6		dochkrshp2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
7		dochkrshp2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
8		dochkrshp2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		dochkrshp2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
10	1 2 3 4 5 6 7 8 9	dochkrshp	⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ) )
11	1 2 3 6 5 7 8 9	dochlkr	⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ 𝑌 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) )
12	10 11	bitrd	⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( 𝐿 ‘ 𝐺 ) ∈ 𝑌 ) ) )