This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 4atexlemv

Description: Lemma for 4atexlem7 . (Contributed by NM, 23-Nov-2012)

Ref Expression
Hypotheses 4thatlem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
4thatlem0.l ⊢ ≤ = ( le ‘ 𝐾 )
4thatlem0.j ⊢ ∨ = ( join ‘ 𝐾 )
4thatlem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
4thatlem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4thatlem0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4thatlem0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
4thatlem0.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
Assertion 4atexlemv ( 𝜑 → 𝑉 ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 4thatlem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
2 4thatlem0.l ⊢ ≤ = ( le ‘ 𝐾 )
3 4thatlem0.j ⊢ ∨ = ( join ‘ 𝐾 )
4 4thatlem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 4thatlem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 4thatlem0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 4thatlem0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 4thatlem0.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
9 1 4atexlemk ⊢ ( 𝜑 → 𝐾 ∈ HL )
10 1 4atexlemw ⊢ ( 𝜑 → 𝑊 ∈ 𝐻 )
11 1 4atexlempw ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
12 1 4atexlems ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
13 1 2 3 5 4atexlempns ⊢ ( 𝜑 → 𝑃 ≠ 𝑆 )
14 2 3 4 5 6 8 lhpat2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ) → 𝑉 ∈ 𝐴 )
15 9 10 11 12 13 14 syl212anc ⊢ ( 𝜑 → 𝑉 ∈ 𝐴 )