Metamath Proof Explorer


Theorem dochsscl

Description: If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015)

Ref Expression
Hypotheses dochsscl.h 𝐻 = ( LHyp ‘ 𝐾 )
dochsscl.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochsscl.v 𝑉 = ( Base ‘ 𝑈 )
dochsscl.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dochsscl.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
dochsscl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
dochsscl.x ( 𝜑𝑋𝑉 )
dochsscl.y ( 𝜑𝑌 ∈ ran 𝐼 )
Assertion dochsscl ( 𝜑 → ( 𝑋𝑌 ↔ ( ‘ ( 𝑋 ) ) ⊆ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 dochsscl.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dochsscl.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dochsscl.v 𝑉 = ( Base ‘ 𝑈 )
4 dochsscl.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5 dochsscl.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
6 dochsscl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
7 dochsscl.x ( 𝜑𝑋𝑉 )
8 dochsscl.y ( 𝜑𝑌 ∈ ran 𝐼 )
9 6 adantr ( ( 𝜑𝑋𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 7 adantr ( ( 𝜑𝑋𝑌 ) → 𝑋𝑉 )
11 1 2 3 5 dochssv ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝑉 ) → ( 𝑋 ) ⊆ 𝑉 )
12 9 10 11 syl2anc ( ( 𝜑𝑋𝑌 ) → ( 𝑋 ) ⊆ 𝑉 )
13 1 2 4 3 dihrnss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → 𝑌𝑉 )
14 6 8 13 syl2anc ( 𝜑𝑌𝑉 )
15 14 adantr ( ( 𝜑𝑋𝑌 ) → 𝑌𝑉 )
16 simpr ( ( 𝜑𝑋𝑌 ) → 𝑋𝑌 )
17 1 2 3 5 dochss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝑉𝑋𝑌 ) → ( 𝑌 ) ⊆ ( 𝑋 ) )
18 9 15 16 17 syl3anc ( ( 𝜑𝑋𝑌 ) → ( 𝑌 ) ⊆ ( 𝑋 ) )
19 1 2 3 5 dochss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋 ) ⊆ 𝑉 ∧ ( 𝑌 ) ⊆ ( 𝑋 ) ) → ( ‘ ( 𝑋 ) ) ⊆ ( ‘ ( 𝑌 ) ) )
20 9 12 18 19 syl3anc ( ( 𝜑𝑋𝑌 ) → ( ‘ ( 𝑋 ) ) ⊆ ( ‘ ( 𝑌 ) ) )
21 8 adantr ( ( 𝜑𝑋𝑌 ) → 𝑌 ∈ ran 𝐼 )
22 1 4 5 dochoc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( ‘ ( 𝑌 ) ) = 𝑌 )
23 9 21 22 syl2anc ( ( 𝜑𝑋𝑌 ) → ( ‘ ( 𝑌 ) ) = 𝑌 )
24 20 23 sseqtrd ( ( 𝜑𝑋𝑌 ) → ( ‘ ( 𝑋 ) ) ⊆ 𝑌 )
25 1 2 3 5 dochocss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝑉 ) → 𝑋 ⊆ ( ‘ ( 𝑋 ) ) )
26 6 7 25 syl2anc ( 𝜑𝑋 ⊆ ( ‘ ( 𝑋 ) ) )
27 sstr ( ( 𝑋 ⊆ ( ‘ ( 𝑋 ) ) ∧ ( ‘ ( 𝑋 ) ) ⊆ 𝑌 ) → 𝑋𝑌 )
28 26 27 sylan ( ( 𝜑 ∧ ( ‘ ( 𝑋 ) ) ⊆ 𝑌 ) → 𝑋𝑌 )
29 24 28 impbida ( 𝜑 → ( 𝑋𝑌 ↔ ( ‘ ( 𝑋 ) ) ⊆ 𝑌 ) )