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Metamath Proof Explorer

Theorem dihmeetlem6

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014)

Ref Expression
Hypotheses dihmeetlem6.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem6.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetlem6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihmeetlem6.j ⊢ ∨ = ( join ‘ 𝐾 )
dihmeetlem6.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetlem6.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion dihmeetlem6 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 )

Proof

Step Hyp Ref Expression
1 dihmeetlem6.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem6.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetlem6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihmeetlem6.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihmeetlem6.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihmeetlem6.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 simprlr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ 𝑄 ≤ 𝑊 )
8 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ HL )
9 8 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ Lat )
10 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 )
11 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑌 ∈ 𝐵 )
12 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
13 9 10 11 12 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
14 simprll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 )
15 1 6 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
16 14 15 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐵 )
17 simpl1r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑊 ∈ 𝐻 )
18 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
19 17 18 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑊 ∈ 𝐵 )
20 1 2 4 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) )
21 9 13 16 19 20 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) )
22 simpr ⊢ ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) → 𝑄 ≤ 𝑊 )
23 21 22 biimtrrdi ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 → 𝑄 ≤ 𝑊 ) )
24 7 23 mtod ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 )
25 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ≤ 𝑋 )
26 1 2 4 5 6 dihmeetlem5 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )
27 8 10 11 14 25 26 syl32anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 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 ∨ 𝑄 ) ) ≤ 𝑊 ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) )
29 24 28 mtbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 )