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Metamath Proof Explorer

Theorem dihmeetlem7N

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

Ref Expression
Hypotheses dihmeetlem7.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem7.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetlem7.j ⊢ ∨ = ( join ‘ 𝐾 )
dihmeetlem7.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetlem7.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion dihmeetlem7N ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ∧ 𝑌 ) = ( 𝑋 ∧ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem7.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem7.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetlem7.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihmeetlem7.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihmeetlem7.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 simprr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ¬ 𝑝 ≤ 𝑌 )
7 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ HL )
8 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
9 7 8 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ AtLat )
10 simprl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝑝 ∈ 𝐴 )
11 simpl3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝑌 ∈ 𝐵 )
12 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
13 1 2 4 12 5 atnle ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑝 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( ¬ 𝑝 ≤ 𝑌 ↔ ( 𝑝 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) )
14 9 10 11 13 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ¬ 𝑝 ≤ 𝑌 ↔ ( 𝑝 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) )
15 6 14 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( 𝑝 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) )
16 15 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑝 ∧ 𝑌 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) )
17 7 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ Lat )
18 simpl2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝑋 ∈ 𝐵 )
19 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
20 17 18 11 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
21 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 )
22 17 18 11 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 )
23 1 2 3 4 5 atmod1i2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑝 ∧ 𝑌 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ∧ 𝑌 ) )
24 7 10 20 11 22 23 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑝 ∧ 𝑌 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ∧ 𝑌 ) )
25 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
26 7 25 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ OL )
27 1 3 12 olj01 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) )
28 26 20 27 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) )
29 16 24 28 3eqtr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ∧ 𝑌 ) = ( 𝑋 ∧ 𝑌 ) )