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Metamath Proof Explorer

Theorem 4atexlemex4

Description: Lemma for 4atexlem7 . Show that when C = S , D satisfies the existence condition of the consequent. (Contributed by NM, 26-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 )
4thatlem0.c ⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
4thatlem0.d ⊢ 𝐷 = ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
Assertion 4atexlemex4 ( ( 𝜑 ∧ 𝐶 = 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )

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 4thatlem0.c ⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
10 4thatlem0.d ⊢ 𝐷 = ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
11 1 2 3 5 7 4atexlemswapqr ⊢ ( 𝜑 → ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) )
12 1 2 3 4 5 6 7 8 9 10 4atexlemcnd ⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
13 pm13.18 ⊢ ( ( 𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷 ) → 𝑆 ≠ 𝐷 )
14 13 necomd ⊢ ( ( 𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷 ) → 𝐷 ≠ 𝑆 )
15 14 expcom ⊢ ( 𝐶 ≠ 𝐷 → ( 𝐶 = 𝑆 → 𝐷 ≠ 𝑆 ) )
16 12 15 syl ⊢ ( 𝜑 → ( 𝐶 = 𝑆 → 𝐷 ≠ 𝑆 ) )
17 biid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) )
18 eqid ⊢ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 )
19 17 2 3 4 5 6 18 8 10 4atexlemex2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ∧ 𝐷 ≠ 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )
20 11 16 19 syl6an ⊢ ( 𝜑 → ( 𝐶 = 𝑆 → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) )
21 20 imp ⊢ ( ( 𝜑 ∧ 𝐶 = 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )