# Metamath Proof Explorer

## Theorem dochlkr

Description: Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014)

Ref Expression
Hypotheses dochlkr.h 𝐻 = ( LHyp ‘ 𝐾 )
dochlkr.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochlkr.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochlkr.f 𝐹 = ( LFnl ‘ 𝑈 )
dochlkr.y 𝑌 = ( LSHyp ‘ 𝑈 )
dochlkr.l 𝐿 = ( LKer ‘ 𝑈 )
dochlkr.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
dochlkr.g ( 𝜑𝐺𝐹 )
Assertion dochlkr ( 𝜑 → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ↔ ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( 𝐿𝐺 ) ∧ ( 𝐿𝐺 ) ∈ 𝑌 ) ) )

### Proof

Step Hyp Ref Expression
1 dochlkr.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dochlkr.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 dochlkr.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 dochlkr.f 𝐹 = ( LFnl ‘ 𝑈 )
5 dochlkr.y 𝑌 = ( LSHyp ‘ 𝑈 )
6 dochlkr.l 𝐿 = ( LKer ‘ 𝑈 )
7 dochlkr.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
8 dochlkr.g ( 𝜑𝐺𝐹 )
9 eqid ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
10 1 3 7 dvhlmod ( 𝜑𝑈 ∈ LMod )
11 9 4 6 10 8 lkrssv ( 𝜑 → ( 𝐿𝐺 ) ⊆ ( Base ‘ 𝑈 ) )
12 1 3 9 2 dochocss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐿𝐺 ) ⊆ ( Base ‘ 𝑈 ) ) → ( 𝐿𝐺 ) ⊆ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) )
13 7 11 12 syl2anc ( 𝜑 → ( 𝐿𝐺 ) ⊆ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) )
14 13 adantr ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( 𝐿𝐺 ) ⊆ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) )
15 1 3 7 dvhlvec ( 𝜑𝑈 ∈ LVec )
16 15 adantr ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → 𝑈 ∈ LVec )
17 10 adantr ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → 𝑈 ∈ LMod )
18 simpr ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 )
19 9 5 17 18 lshpne ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ≠ ( Base ‘ 𝑈 ) )
20 19 ex ( 𝜑 → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ≠ ( Base ‘ 𝑈 ) ) )
21 9 5 4 6 15 8 lkrshpor ( 𝜑 → ( ( 𝐿𝐺 ) ∈ 𝑌 ∨ ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) ) )
22 21 ord ( 𝜑 → ( ¬ ( 𝐿𝐺 ) ∈ 𝑌 → ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) ) )
23 2fveq3 ( ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( ‘ ( ‘ ( Base ‘ 𝑈 ) ) ) )
24 23 adantl ( ( 𝜑 ∧ ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( ‘ ( ‘ ( Base ‘ 𝑈 ) ) ) )
25 7 adantr ( ( 𝜑 ∧ ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
26 1 3 2 9 25 dochoc1 ( ( 𝜑 ∧ ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) ) → ( ‘ ( ‘ ( Base ‘ 𝑈 ) ) ) = ( Base ‘ 𝑈 ) )
27 24 26 eqtrd ( ( 𝜑 ∧ ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( Base ‘ 𝑈 ) )
28 27 ex ( 𝜑 → ( ( 𝐿𝐺 ) = ( Base ‘ 𝑈 ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( Base ‘ 𝑈 ) ) )
29 22 28 syld ( 𝜑 → ( ¬ ( 𝐿𝐺 ) ∈ 𝑌 → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( Base ‘ 𝑈 ) ) )
30 29 necon1ad ( 𝜑 → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ≠ ( Base ‘ 𝑈 ) → ( 𝐿𝐺 ) ∈ 𝑌 ) )
31 20 30 syld ( 𝜑 → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 → ( 𝐿𝐺 ) ∈ 𝑌 ) )
32 31 imp ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( 𝐿𝐺 ) ∈ 𝑌 )
33 5 16 32 18 lshpcmp ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( ( 𝐿𝐺 ) ⊆ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ↔ ( 𝐿𝐺 ) = ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ) )
34 14 33 mpbid ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( 𝐿𝐺 ) = ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) )
35 34 eqcomd ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( 𝐿𝐺 ) )
36 35 32 jca ( ( 𝜑 ∧ ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ) → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( 𝐿𝐺 ) ∧ ( 𝐿𝐺 ) ∈ 𝑌 ) )
37 36 ex ( 𝜑 → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( 𝐿𝐺 ) ∧ ( 𝐿𝐺 ) ∈ 𝑌 ) ) )
38 eleq1 ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( 𝐿𝐺 ) → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ↔ ( 𝐿𝐺 ) ∈ 𝑌 ) )
39 38 biimpar ( ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( 𝐿𝐺 ) ∧ ( 𝐿𝐺 ) ∈ 𝑌 ) → ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 )
40 37 39 impbid1 ( 𝜑 → ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) ∈ 𝑌 ↔ ( ( ‘ ( ‘ ( 𝐿𝐺 ) ) ) = ( 𝐿𝐺 ) ∧ ( 𝐿𝐺 ) ∈ 𝑌 ) ) )