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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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) )

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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) )
11 5 6 10 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) )
12 simp1r ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 )
13 simpl2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑋 ≤ 𝑊 )
14 13 3ad2ant1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑊 )
15 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝐾 ∈ HL )
16 15 3ad2ant1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ HL )
17 16 hllatd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ Lat )
18 simpl2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑋 ∈ 𝐵 )
19 18 3ad2ant1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ∈ 𝐵 )
20 simpl3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑌 ∈ 𝐵 )
21 20 3ad2ant1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑌 ∈ 𝐵 )
22 simpl1r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑊 ∈ 𝐻 )
23 22 3ad2ant1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑊 ∈ 𝐻 )
24 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
25 23 24 syl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑊 ∈ 𝐵 )
26 1 2 8 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊 ) ↔ 𝑋 ≤ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) )
27 17 19 21 25 26 syl13anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊 ) ↔ 𝑋 ≤ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) )
28 12 14 27 mpbi2and ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
30 simp1l2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
31 1 8 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
32 17 21 25 31 syl3anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 )
34 17 21 25 33 syl3anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
47 29 44 46 syl2anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
57 29 30 56 syl2anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
58 simp1l3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) )
59 simp3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
61 29 58 44 59 60 syl112anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
62 55 57 61 3sstr4d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) )
63 62 3exp ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) → ( ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) )
64 63 expd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) → ( ¬ 𝑟 ≤ 𝑊 → ( ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) ) )
65 64 imp4a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) → ( ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) )
66 65 rexlimdv ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) )
67 11 66 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) )