Metamath Proof Explorer

Theorem dihmeetlem20N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem14.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem14.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetlem14.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihmeetlem14.j ⊢ ∨ = ( join ‘ 𝐾 )
dihmeetlem14.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetlem14.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihmeetlem14.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihmeetlem14.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihmeetlem14.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihmeetlem20N ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem14.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem14.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetlem14.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihmeetlem14.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihmeetlem14.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihmeetlem14.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihmeetlem14.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihmeetlem14.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihmeetlem14.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
12 simp3ll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
13 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 )
14 1 2 4 5 6 3 lhpmcvr6N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) )
15 10 11 12 13 14 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) )
16 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) )
17 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
18 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ HL )
19 18 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
20 1 5 latmcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑋 ) = ( 𝑋 ∧ 𝑌 ) )
21 19 12 17 20 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑌 ∧ 𝑋 ) = ( 𝑋 ∧ 𝑌 ) )
22 21 13 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑌 ∧ 𝑋 ) ≤ 𝑊 )
23 1 2 4 5 6 3 lhpmcvr6N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑋 ) ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) )
24 10 16 17 22 23 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) )
25 reeanv ⊢ ( ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ↔ ( ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) )
26 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
27 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
28 12 3ad2ant1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → 𝑌 ∈ 𝐵 )
29 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → 𝑞 ∈ 𝐴 )
30 simp3l1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ¬ 𝑞 ≤ 𝑊 )
31 29 30 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) )
32 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → 𝑟 ∈ 𝐴 )
33 simp3r1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ¬ 𝑟 ≤ 𝑊 )
34 32 33 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) )
35 simp3l3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → 𝑞 ≤ 𝑋 )
36 simp3r3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → 𝑟 ≤ 𝑌 )
37 simp13r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 )
38 35 36 37 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ( 𝑞 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) )
39 1 2 3 4 5 6 7 8 9 dihmeetlem19N ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑞 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
40 26 27 28 31 34 38 39 syl33anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
41 40 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) → ( ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) ) )
42 41 rexlimdvv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) )
43 25 42 biimtrrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑌 ∧ 𝑞 ≤ 𝑋 ) ∧ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) )
44 15 24 43 mp2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )