Theorem lcfl5a

Description: Property of a functional with a closed kernel. TODO: Make lcfl5 etc. obsolete and rewrite without C hypothesis? (Contributed by NM, 29-Jan-2015)

	Ref	Expression
Hypotheses	lcfl5a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfl5a.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl5a.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl5a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl5a.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lcfl5a.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfl5a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfl5a.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion	lcfl5a	⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ↔ ( 𝐿 ‘ 𝐺 ) ∈ ran 𝐼 ) )

Proof

Step	Hyp	Ref	Expression
1		lcfl5a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfl5a.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfl5a.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfl5a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		lcfl5a.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		lcfl5a.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		lcfl5a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		lcfl5a.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
9		eqid	⊢ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
10	9 8	lcfl1	⊢ ( 𝜑 → ( 𝐺 ∈ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) )
11	1 2 3 4 5 6 9 7 8	lcfl5	⊢ ( 𝜑 → ( 𝐺 ∈ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ↔ ( 𝐿 ‘ 𝐺 ) ∈ ran 𝐼 ) )
12	10 11	bitr3d	⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ↔ ( 𝐿 ‘ 𝐺 ) ∈ ran 𝐼 ) )