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Metamath Proof Explorer

Theorem dihord5apre

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014)

Ref Expression
Hypotheses dihord5apre.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihord5apre.l ⊢ ≤ = ( le ‘ 𝐾 )
dihord5apre.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihord5apre.j ⊢ ∨ = ( join ‘ 𝐾 )
dihord5apre.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihord5apre.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihord5apre.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihord5apre.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihord5apre.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihord5apre ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )

Proof

Step Hyp Ref Expression
1 dihord5apre.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihord5apre.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihord5apre.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihord5apre.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihord5apre.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihord5apre.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihord5apre.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihord5apre.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihord5apre.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) )
12 1 2 4 5 6 3 lhpmcvr2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) )
13 10 11 12 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) )
14 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝐾 ∈ HL )
15 14 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝐾 ∈ Lat )
16 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑋 ∈ 𝐵 )
17 simp3ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑟 ∈ 𝐴 )
18 1 6 atbase ⊢ ( 𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵 )
19 17 18 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑟 ∈ 𝐵 )
20 1 4 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 )
21 15 19 16 20 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 )
22 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑌 ∈ 𝐵 )
23 1 2 4 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ ( 𝑟 ∨ 𝑋 ) )
24 15 19 16 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑋 ≤ ( 𝑟 ∨ 𝑋 ) )
25 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
26 simp3lr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ¬ 𝑟 ≤ 𝑊 )
27 1 2 4 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) )
28 15 19 16 27 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) )
29 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑊 ∈ 𝐻 )
30 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
31 29 30 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑊 ∈ 𝐵 )
32 1 2 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑟 ∈ 𝐵 ∧ ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) ∧ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) → 𝑟 ≤ 𝑊 ) )
33 15 19 21 31 32 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) ∧ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) → 𝑟 ≤ 𝑊 ) )
34 28 33 mpand ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 → 𝑟 ≤ 𝑊 ) )
35 26 34 mtod ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ¬ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 )
36 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) )
37 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
38 1 2 4 5 6 3 lhple ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) = 𝑋 )
39 25 36 37 38 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) = 𝑋 )
40 39 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( 𝑟 ∨ 𝑋 ) )
41 eqid ⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
42 eqid ⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
43 1 2 4 5 6 3 9 41 42 7 8 dihvalcq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( 𝑟 ∨ 𝑋 ) ) ) → ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) )
44 25 21 35 36 40 43 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) )
45 3 7 25 dvhlmod ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑈 ∈ LMod )
46 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
47 46 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
48 45 47 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
49 2 6 3 7 42 46 diclss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ 𝑈 ) )
50 25 36 49 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ 𝑈 ) )
51 48 50 sseldd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) )
52 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
53 15 22 31 52 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
54 1 2 5 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 )
55 15 22 31 54 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 )
56 1 2 3 7 41 46 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
57 25 53 55 56 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
58 48 57 sseldd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
59 8 lsmub1 ⊢ ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) )
60 51 58 59 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) )
61 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) )
62 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 )
63 1 2 4 5 6 3 9 41 42 7 8 dihvalcq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) )
64 25 61 36 62 63 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) )
65 60 64 sseqtrrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼 ‘ 𝑌 ) )
66 39 fveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
67 1 2 3 9 41 dihvalb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
68 25 37 67 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
69 66 68 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( 𝐼 ‘ 𝑋 ) )
70 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) )
71 69 70 eqsstrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) )
72 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ∈ 𝐵 )
73 15 21 31 72 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ∈ 𝐵 )
74 1 2 5 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ≤ 𝑊 )
75 15 21 31 74 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ≤ 𝑊 )
76 1 2 3 7 41 46 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ∈ 𝐵 ∧ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
77 25 73 75 76 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
78 48 77 sseldd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
79 1 3 9 7 46 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
80 25 22 79 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
81 48 80 sseldd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) )
82 8 lsmlub ⊢ ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ↔ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) )
83 51 78 81 82 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ↔ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) )
84 65 71 83 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ⊆ ( 𝐼 ‘ 𝑌 ) )
85 44 84 eqsstrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) )
86 1 2 3 9 dihord4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑟 ∨ 𝑋 ) ≤ 𝑌 ) )
87 25 21 35 61 86 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑟 ∨ 𝑋 ) ≤ 𝑌 ) )
88 85 87 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ 𝑋 ) ≤ 𝑌 )
89 1 2 15 16 21 22 24 88 lattrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑋 ≤ 𝑌 )
90 89 3expia ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 ) )
91 90 exp4c ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝑟 ∈ 𝐴 → ( ¬ 𝑟 ≤ 𝑊 → ( ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 → 𝑋 ≤ 𝑌 ) ) ) )
92 91 imp4a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝑟 ∈ 𝐴 → ( ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 ) ) )
93 92 rexlimdv ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 ) )
94 13 93 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 )