Theorem lcfl5

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

	Ref	Expression
Hypotheses	lcfl5.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfl5.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl5.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl5.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfl5.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lcfl5.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfl5.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
	lcfl5.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfl5.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion	lcfl5	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( 𝐿 ‘ 𝐺 ) ∈ ran 𝐼 ) )

Proof

Step	Hyp	Ref	Expression
1		lcfl5.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfl5.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfl5.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfl5.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		lcfl5.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		lcfl5.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		lcfl5.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
8		lcfl5.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		lcfl5.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
10	7 9	lcfl1	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) )
11		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
12	1 4 8	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
13	11 5 6 12 9	lkrssv	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ ( Base ‘ 𝑈 ) )
14	1 2 4 11 3 8 13	dochoccl	⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) ∈ ran 𝐼 ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) )
15	10 14	bitr4d	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( 𝐿 ‘ 𝐺 ) ∈ ran 𝐼 ) )