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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → 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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
11 simpl3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) )
12 1 2 4 5 6 3 lhpmcvr2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) → ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) )
13 10 11 12 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) )
14 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝐾 ∈ HL )
15 14 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝐾 ∈ Lat )
16 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑋𝐵 )
17 simp3ll ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑟𝐴 )
18 1 6 atbase ( 𝑟𝐴𝑟𝐵 )
19 17 18 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑟𝐵 )
20 1 4 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑟𝐵𝑋𝐵 ) → ( 𝑟 𝑋 ) ∈ 𝐵 )
21 15 19 16 20 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 𝑋 ) ∈ 𝐵 )
22 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑌𝐵 )
23 1 2 4 latlej2 ( ( 𝐾 ∈ Lat ∧ 𝑟𝐵𝑋𝐵 ) → 𝑋 ( 𝑟 𝑋 ) )
24 15 19 16 23 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑋 ( 𝑟 𝑋 ) )
25 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
26 simp3lr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ¬ 𝑟 𝑊 )
27 1 2 4 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑟𝐵𝑋𝐵 ) → 𝑟 ( 𝑟 𝑋 ) )
28 15 19 16 27 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑟 ( 𝑟 𝑋 ) )
29 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑊𝐻 )
30 1 3 lhpbase ( 𝑊𝐻𝑊𝐵 )
31 29 30 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑊𝐵 )
32 1 2 lattr ( ( 𝐾 ∈ Lat ∧ ( 𝑟𝐵 ∧ ( 𝑟 𝑋 ) ∈ 𝐵𝑊𝐵 ) ) → ( ( 𝑟 ( 𝑟 𝑋 ) ∧ ( 𝑟 𝑋 ) 𝑊 ) → 𝑟 𝑊 ) )
33 15 19 21 31 32 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ( 𝑟 𝑋 ) ∧ ( 𝑟 𝑋 ) 𝑊 ) → 𝑟 𝑊 ) )
34 28 33 mpand ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 𝑋 ) 𝑊𝑟 𝑊 ) )
35 26 34 mtod ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ¬ ( 𝑟 𝑋 ) 𝑊 )
36 simp3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) )
37 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝑋𝐵𝑋 𝑊 ) )
38 1 2 4 5 6 3 lhple ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ( 𝑟 𝑋 ) 𝑊 ) = 𝑋 )
39 25 36 37 38 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 𝑋 ) 𝑊 ) = 𝑋 )
40 39 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑟 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝑟 𝑋 ) 𝑊 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
49 2 6 3 7 42 46 diclss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ 𝑈 ) )
50 25 36 49 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ 𝑈 ) )
51 48 50 sseldd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) )
52 1 5 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵 ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
53 15 22 31 52 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
54 1 2 5 latmle2 ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵 ) → ( 𝑌 𝑊 ) 𝑊 )
55 15 22 31 54 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 𝑊 ) 𝑊 )
56 1 2 3 7 41 46 diblss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑌 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 𝑊 ) 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
57 25 53 55 56 syl12anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
58 48 57 sseldd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 𝑊 ) ) ) )
61 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) )
62 simp3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐼𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 𝑊 ) ) ) )
64 25 61 36 62 63 syl112anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐼𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 𝑊 ) ) ) )
65 60 64 sseqtrrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼𝑌 ) )
66 39 fveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐼𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
68 25 37 67 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐼𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
69 66 68 eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) = ( 𝐼𝑋 ) )
70 simp2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) )
71 69 70 eqsstrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ⊆ ( 𝐼𝑌 ) )
72 1 5 latmcl ( ( 𝐾 ∈ Lat ∧ ( 𝑟 𝑋 ) ∈ 𝐵𝑊𝐵 ) → ( ( 𝑟 𝑋 ) 𝑊 ) ∈ 𝐵 )
73 15 21 31 72 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 𝑋 ) 𝑊 ) ∈ 𝐵 )
74 1 2 5 latmle2 ( ( 𝐾 ∈ Lat ∧ ( 𝑟 𝑋 ) ∈ 𝐵𝑊𝐵 ) → ( ( 𝑟 𝑋 ) 𝑊 ) 𝑊 )
75 15 21 31 74 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 𝑋 ) 𝑊 ) 𝑊 )
76 1 2 3 7 41 46 diblss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( ( 𝑟 𝑋 ) 𝑊 ) ∈ 𝐵 ∧ ( ( 𝑟 𝑋 ) 𝑊 ) 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
77 25 73 75 76 syl12anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
78 48 77 sseldd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
79 1 3 9 7 46 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ) → ( 𝐼𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
80 25 22 79 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐼𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) )
81 48 80 sseldd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐼𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) )
82 8 lsmlub ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ⊆ ( 𝐼𝑌 ) ) ↔ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ) ⊆ ( 𝐼𝑌 ) ) )
83 51 78 81 82 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ⊆ ( 𝐼𝑌 ) ) ↔ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ) ⊆ ( 𝐼𝑌 ) ) )
84 65 71 83 mpbi2and ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 𝑋 ) 𝑊 ) ) ) ⊆ ( 𝐼𝑌 ) )
85 44 84 eqsstrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ ( 𝑟 𝑋 ) ) ⊆ ( 𝐼𝑌 ) )
86 1 2 3 9 dihord4 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑟 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝑟 𝑋 ) 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑟 𝑋 ) ) ⊆ ( 𝐼𝑌 ) ↔ ( 𝑟 𝑋 ) 𝑌 ) )
87 25 21 35 61 86 syl121anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( ( 𝐼 ‘ ( 𝑟 𝑋 ) ) ⊆ ( 𝐼𝑌 ) ↔ ( 𝑟 𝑋 ) 𝑌 ) )
88 85 87 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 𝑋 ) 𝑌 )
89 1 2 15 16 21 22 24 88 lattrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) ) → 𝑋 𝑌 )
90 89 3expia ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → ( ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) → 𝑋 𝑌 ) )
91 90 exp4c ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → ( 𝑟𝐴 → ( ¬ 𝑟 𝑊 → ( ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌𝑋 𝑌 ) ) ) )
92 91 imp4a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → ( 𝑟𝐴 → ( ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) → 𝑋 𝑌 ) ) )
93 92 rexlimdv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → ( ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( 𝑌 𝑊 ) ) = 𝑌 ) → 𝑋 𝑌 ) )
94 13 93 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) → 𝑋 𝑌 )