Metamath Proof Explorer

Theorem dalem1

Description: Lemma for dath . Show the lines P S and Q T are different. (Contributed by NM, 9-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 )
dalem1.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem1.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
Assertion dalem1 ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ≠ ( 𝑄 ∨ 𝑇 ) )

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 dalem1.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalem1.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7 1 dalemclpjs ⊢ ( 𝜑 → 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) )
8 1 dalem-clpjq ⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) )
9 8 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) )
10 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
11 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
12 1 dalemsea ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
13 2 3 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑆 ) )
14 10 11 12 13 syl3anc ⊢ ( 𝜑 → 𝑃 ≤ ( 𝑃 ∨ 𝑆 ) )
15 14 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑆 ) )
16 1 dalemqea ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
17 1 dalemtea ⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
18 2 3 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) → 𝑄 ≤ ( 𝑄 ∨ 𝑇 ) )
19 10 16 17 18 syl3anc ⊢ ( 𝜑 → 𝑄 ≤ ( 𝑄 ∨ 𝑇 ) )
20 19 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → 𝑄 ≤ ( 𝑄 ∨ 𝑇 ) )
21 simpr ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) )
22 20 21 breqtrrd ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) )
23 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
24 1 4 dalempeb ⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
25 1 4 dalemqeb ⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
26 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
27 26 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
28 10 11 12 27 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
29 26 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) ) ↔ ( 𝑃 ∨ 𝑄 ) ≤ ( 𝑃 ∨ 𝑆 ) ) )
30 23 24 25 28 29 syl13anc ⊢ ( 𝜑 → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) ) ↔ ( 𝑃 ∨ 𝑄 ) ≤ ( 𝑃 ∨ 𝑆 ) ) )
31 30 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) ) ↔ ( 𝑃 ∨ 𝑄 ) ≤ ( 𝑃 ∨ 𝑆 ) ) )
32 15 22 31 mpbi2and ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ( 𝑃 ∨ 𝑄 ) ≤ ( 𝑃 ∨ 𝑆 ) )
33 1 dalemrea ⊢ ( 𝜑 → 𝑅 ∈ 𝐴 )
34 1 dalemyeo ⊢ ( 𝜑 → 𝑌 ∈ 𝑂 )
35 3 4 5 6 lplnri1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑂 ) → 𝑃 ≠ 𝑄 )
36 10 11 16 33 34 35 syl131anc ⊢ ( 𝜑 → 𝑃 ≠ 𝑄 )
37 2 3 4 ps-1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ≤ ( 𝑃 ∨ 𝑆 ) ↔ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑆 ) ) )
38 10 11 16 36 11 12 37 syl132anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ≤ ( 𝑃 ∨ 𝑆 ) ↔ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑆 ) ) )
39 38 adantr ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ( ( 𝑃 ∨ 𝑄 ) ≤ ( 𝑃 ∨ 𝑆 ) ↔ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑆 ) ) )
40 32 39 mpbid ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑆 ) )
41 40 breq2d ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ( 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ) )
42 9 41 mtbid ⊢ ( ( 𝜑 ∧ ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) ) → ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) )
43 42 ex ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) = ( 𝑄 ∨ 𝑇 ) → ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ) )
44 43 necon2ad ⊢ ( 𝜑 → ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) → ( 𝑃 ∨ 𝑆 ) ≠ ( 𝑄 ∨ 𝑇 ) ) )
45 7 44 mpd ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ≠ ( 𝑄 ∨ 𝑇 ) )