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Metamath Proof Explorer

Theorem dochss

Description: Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014)

Ref Expression
Hypotheses dochss.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dochss.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dochss.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
dochss.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dochss ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 dochss.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dochss.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dochss.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 dochss.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
5 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → 𝐾 ∈ HL )
6 hlclat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat )
7 5 6 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → 𝐾 ∈ CLat )
8 ssrab2 ⊢ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ ( Base ‘ 𝐾 )
9 8 a1i ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ ( Base ‘ 𝐾 ) )
10 simpll3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) ∧ 𝑧 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) ) → 𝑋 ⊆ 𝑌 )
11 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) ∧ 𝑧 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) ) → 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) )
12 10 11 sstrd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) ∧ 𝑧 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) ) → 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) )
13 12 ex ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) ∧ 𝑧 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) → 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) ) )
14 13 ss2rabdv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } )
15 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
17 eqid ⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 )
18 15 16 17 clatglbss ⊢ ( ( 𝐾 ∈ CLat ∧ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ ( Base ‘ 𝐾 ) ∧ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ( le ‘ 𝐾 ) ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) )
19 7 9 14 18 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ( le ‘ 𝐾 ) ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) )
20 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
21 5 20 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → 𝐾 ∈ OP )
22 15 17 clatglbcl ⊢ ( ( 𝐾 ∈ CLat ∧ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ ( Base ‘ 𝐾 ) ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) )
23 7 8 22 sylancl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) )
24 ssrab2 ⊢ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ ( Base ‘ 𝐾 )
25 15 17 clatglbcl ⊢ ( ( 𝐾 ∈ CLat ∧ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ⊆ ( Base ‘ 𝐾 ) ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) )
26 7 24 25 sylancl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) )
27 eqid ⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
28 15 16 27 oplecon3b ⊢ ( ( 𝐾 ∈ OP ∧ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ( le ‘ 𝐾 ) ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ↔ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
29 21 23 26 28 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ( le ‘ 𝐾 ) ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ↔ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
30 19 29 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) )
31 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
32 15 27 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ∈ ( Base ‘ 𝐾 ) )
33 21 26 32 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ∈ ( Base ‘ 𝐾 ) )
34 15 27 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ∈ ( Base ‘ 𝐾 ) )
35 21 23 34 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ∈ ( Base ‘ 𝐾 ) )
36 eqid ⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
37 15 16 1 36 dihord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) ↔ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
38 31 33 35 37 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) ↔ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
39 30 38 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
40 15 17 27 1 36 2 3 4 dochval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑌 ) = ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
41 40 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑌 ) = ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑌 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
42 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → 𝑋 ⊆ 𝑌 )
43 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → 𝑌 ⊆ 𝑉 )
44 42 43 sstrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → 𝑋 ⊆ 𝑉 )
45 15 17 27 1 36 2 3 4 dochval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑋 ) = ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
46 31 44 45 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑋 ) = ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑧 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑧 ) } ) ) ) )
47 39 41 46 3sstr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) )