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Metamath Proof Explorer

Theorem islvol2

Description: The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012)

Ref Expression
Hypotheses islvol5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
islvol5.l ⊢ ≤ = ( le ‘ 𝐾 )
islvol5.j ⊢ ∨ = ( join ‘ 𝐾 )
islvol5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
islvol5.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
Assertion islvol2 ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑉 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ ¬ 𝑠 ≤ ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ∧ 𝑋 = ( ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ∨ 𝑠 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 islvol5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 islvol5.l ⊢ ≤ = ( le ‘ 𝐾 )
3 islvol5.j ⊢ ∨ = ( join ‘ 𝐾 )
4 islvol5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 islvol5.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
6 1 5 lvolbase ⊢ ( 𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵 )
7 6 pm4.71ri ⊢ ( 𝑋 ∈ 𝑉 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) )
8 1 2 3 4 5 islvol5 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑉 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ ¬ 𝑠 ≤ ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ∧ 𝑋 = ( ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ∨ 𝑠 ) ) ) )
9 8 pm5.32da ⊢ ( 𝐾 ∈ HL → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ ¬ 𝑠 ≤ ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ∧ 𝑋 = ( ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ∨ 𝑠 ) ) ) ) )
10 7 9 bitrid ⊢ ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑉 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ ¬ 𝑠 ≤ ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ∧ 𝑋 = ( ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ∨ 𝑠 ) ) ) ) )