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Metamath Proof Explorer

Theorem dihjatc1

Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change .\/ order of ( X ./\ Y ) .\/ Q here and down? (Contributed by NM, 6-Apr-2014)

Ref Expression
Hypotheses dihjatc1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihjatc1.l ⊢ ≤ = ( le ‘ 𝐾 )
dihjatc1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjatc1.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjatc1.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihjatc1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjatc1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjatc1.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjatc1.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihjatc1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjatc1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihjatc1.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihjatc1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihjatc1.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihjatc1.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihjatc1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihjatc1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihjatc1.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihjatc1.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ HL )
12 11 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
13 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
14 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
15 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
16 12 13 14 15 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
17 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ∈ 𝐴 )
18 1 6 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
19 17 18 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ∈ 𝐵 )
20 1 4 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∈ 𝐵 )
21 12 16 19 20 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∈ 𝐵 )
22 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
23 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ≤ 𝑋 )
24 1 2 3 4 5 6 dihmeetlem6 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 )
25 10 13 14 22 23 24 syl32anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ¬ ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 )
26 1 2 4 5 6 dihmeetlem5 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )
27 11 13 14 17 23 26 syl32anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )
28 27 breq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) )
29 25 28 mtbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ¬ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 )
30 1 2 4 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → 𝑄 ≤ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )
31 12 16 19 30 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ≤ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )
32 1 2 4 5 6 3 9 7 8 dihvalcq2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∈ 𝐵 ∧ ¬ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
33 10 21 29 22 31 32 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
34 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
35 2 5 34 6 3 lhpmat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑄 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) )
36 10 22 35 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑄 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) )
37 36 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑄 ∧ 𝑊 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) )
38 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 )
39 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
40 38 39 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑊 ∈ 𝐵 )
41 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 )
42 1 2 4 5 6 atmod1i2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑄 ∧ 𝑊 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) )
43 11 17 16 40 41 42 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑄 ∧ 𝑊 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) )
44 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
45 11 44 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ OL )
46 1 4 34 olj01 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) )
47 45 16 46 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) )
48 37 43 47 3eqtr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) = ( 𝑋 ∧ 𝑌 ) )
49 48 fveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) = ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) )
50 49 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )
51 33 50 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )