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

Metamath Proof Explorer

Theorem 3dimlem1

Description: Lemma for 3dim1 . (Contributed by NM, 25-Jul-2012)

Ref Expression
Hypotheses 3dim0.j ⊢ ∨ = ( join ‘ 𝐾 )
3dim0.l ⊢ ≤ = ( le ‘ 𝐾 )
3dim0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion 3dimlem1 ( ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) ∧ 𝑃 = 𝑄 ) → ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ) )

Proof

Step Hyp Ref Expression
1 3dim0.j ⊢ ∨ = ( join ‘ 𝐾 )
2 3dim0.l ⊢ ≤ = ( le ‘ 𝐾 )
3 3dim0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 neeq1 ⊢ ( 𝑃 = 𝑄 → ( 𝑃 ≠ 𝑅 ↔ 𝑄 ≠ 𝑅 ) )
5 oveq1 ⊢ ( 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) )
6 5 breq2d ⊢ ( 𝑃 = 𝑄 → ( 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ↔ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) )
7 6 notbid ⊢ ( 𝑃 = 𝑄 → ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ↔ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) )
8 5 oveq1d ⊢ ( 𝑃 = 𝑄 → ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) )
9 8 breq2d ⊢ ( 𝑃 = 𝑄 → ( 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ↔ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) )
10 9 notbid ⊢ ( 𝑃 = 𝑄 → ( ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ↔ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) )
11 4 7 10 3anbi123d ⊢ ( 𝑃 = 𝑄 → ( ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ) ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) ) )
12 11 biimparc ⊢ ( ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) ∧ 𝑃 = 𝑄 ) → ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ) )