Theorem lcfl2

Description: Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015)

	Ref	Expression
Hypotheses	lcfl2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfl2.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lcfl2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lcfl2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfl2.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
	lcfl2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfl2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion	lcfl2	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ∨ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lcfl2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfl2.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfl2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfl2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		lcfl2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		lcfl2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		lcfl2.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
8		lcfl2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		lcfl2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
10	7 9	lcfl1	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) )
11	1 2 3 4 5 6 8 9	dochkrshp4	⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ∨ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) ) )
12	10 11	bitrd	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ∨ ( 𝐿 ‘ 𝐺 ) = 𝑉 ) ) )