Metamath Proof Explorer


Theorem dihord5b

Description: Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine with other way to have one lhpmcvr2 . (Contributed by NM, 7-Mar-2014)

Ref Expression
Hypotheses dihord3.b 𝐵 = ( Base ‘ 𝐾 )
dihord3.l = ( le ‘ 𝐾 )
dihord3.h 𝐻 = ( LHyp ‘ 𝐾 )
dihord3.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihord5b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) )

Proof

Step Hyp Ref Expression
1 dihord3.b 𝐵 = ( Base ‘ 𝐾 )
2 dihord3.l = ( le ‘ 𝐾 )
3 dihord3.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dihord3.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
6 simpl3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) )
7 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
9 eqid ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
10 1 2 7 8 9 3 lhpmcvr2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) → ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) )
11 5 6 10 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) )
12 simp1r ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 𝑌 )
13 simpl2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → 𝑋 𝑊 )
14 13 3ad2ant1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 𝑊 )
15 simpl1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → 𝐾 ∈ HL )
16 15 3ad2ant1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ HL )
17 16 hllatd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ Lat )
18 simpl2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → 𝑋𝐵 )
19 18 3ad2ant1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋𝐵 )
20 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → 𝑌𝐵 )
21 20 3ad2ant1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑌𝐵 )
22 simpl1r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → 𝑊𝐻 )
23 22 3ad2ant1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑊𝐻 )
24 1 3 lhpbase ( 𝑊𝐻𝑊𝐵 )
25 23 24 syl ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑊𝐵 )
26 1 2 8 latlem12 ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵𝑊𝐵 ) ) → ( ( 𝑋 𝑌𝑋 𝑊 ) ↔ 𝑋 ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) )
27 17 19 21 25 26 syl13anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( 𝑋 𝑌𝑋 𝑊 ) ↔ 𝑋 ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) )
28 12 14 27 mpbi2and ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) )
29 simp1l1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
30 simp1l2 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑋𝐵𝑋 𝑊 ) )
31 1 8 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
32 17 21 25 31 syl3anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
33 1 2 8 latmle2 ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) 𝑊 )
34 17 21 25 33 syl3anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) 𝑊 )
35 eqid ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
36 1 2 3 35 dibord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) 𝑊 ) ) → ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ↔ 𝑋 ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) )
37 29 30 32 34 36 syl112anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ↔ 𝑋 ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) )
38 28 37 mpbird ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) )
39 eqid ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
40 3 39 29 dvhlmod ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod )
41 eqid ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
42 41 lsssssubg ( ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod → ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⊆ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
43 40 42 syl ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⊆ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
44 simp2 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) )
45 eqid ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
46 2 9 3 39 45 41 diclss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
47 29 44 46 syl2anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
48 43 47 sseldd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
49 1 2 3 39 35 41 diblss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
50 29 32 34 49 syl12anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
51 43 50 sseldd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
52 eqid ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
53 52 lsmub2 ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
54 48 51 53 syl2anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
55 38 54 sstrd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
56 1 2 3 4 35 dihvalb ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐼𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
57 29 30 56 syl2anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
58 simp1l3 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) )
59 simp3 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 )
60 1 2 7 8 9 3 4 35 45 39 52 dihvalcq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ∧ ( ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) ) → ( 𝐼𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
61 29 58 44 59 60 syl112anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
62 55 57 61 3sstr4d ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) )
63 62 3exp ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 𝑊 ) → ( ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) ) )
64 63 expd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) → ( ¬ 𝑟 𝑊 → ( ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) ) ) )
65 64 imp4a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) → ( ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) ) )
66 65 rexlimdv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) ) )
67 11 66 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵 ∧ ¬ 𝑌 𝑊 ) ) ∧ 𝑋 𝑌 ) → ( 𝐼𝑋 ) ⊆ ( 𝐼𝑌 ) )