Metamath Proof Explorer

Theorem dalempnes

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012)

Ref Expression
Hypotheses dalema.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) )
dalemc.l ⊢ ≤ = ( le ‘ 𝐾 )
dalemc.j ⊢ ∨ = ( join ‘ 𝐾 )
dalemc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dalempnes.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalempnes.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
Assertion dalempnes ( 𝜑 → 𝑃 ≠ 𝑆 )

Proof

Step Hyp Ref Expression
1 dalema.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) )
2 dalemc.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dalemc.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dalemc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 dalempnes.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalempnes.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
8 1 4 dalemceb ⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) )
9 1 4 dalemseb ⊢ ( 𝜑 → 𝑆 ∈ ( Base ‘ 𝐾 ) )
10 1 4 dalemteb ⊢ ( 𝜑 → 𝑇 ∈ ( Base ‘ 𝐾 ) )
11 simp321 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) → ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) )
12 1 11 sylbi ⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) )
13 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 2 3 latnlej2l ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑇 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ) → ¬ 𝐶 ≤ 𝑆 )
15 7 8 9 10 12 14 syl131anc ⊢ ( 𝜑 → ¬ 𝐶 ≤ 𝑆 )
16 1 dalemclpjs ⊢ ( 𝜑 → 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) )
17 oveq1 ⊢ ( 𝑃 = 𝑆 → ( 𝑃 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑆 ) )
18 17 breq2d ⊢ ( 𝑃 = 𝑆 → ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ↔ 𝐶 ≤ ( 𝑆 ∨ 𝑆 ) ) )
19 16 18 syl5ibcom ⊢ ( 𝜑 → ( 𝑃 = 𝑆 → 𝐶 ≤ ( 𝑆 ∨ 𝑆 ) ) )
20 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
21 1 dalemsea ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
22 3 4 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ) → ( 𝑆 ∨ 𝑆 ) = 𝑆 )
23 20 21 22 syl2anc ⊢ ( 𝜑 → ( 𝑆 ∨ 𝑆 ) = 𝑆 )
24 23 breq2d ⊢ ( 𝜑 → ( 𝐶 ≤ ( 𝑆 ∨ 𝑆 ) ↔ 𝐶 ≤ 𝑆 ) )
25 19 24 sylibd ⊢ ( 𝜑 → ( 𝑃 = 𝑆 → 𝐶 ≤ 𝑆 ) )
26 25 necon3bd ⊢ ( 𝜑 → ( ¬ 𝐶 ≤ 𝑆 → 𝑃 ≠ 𝑆 ) )
27 15 26 mpd ⊢ ( 𝜑 → 𝑃 ≠ 𝑆 )