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	syl5bb	⊢ ( 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 ) ) ) ) )