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Metamath Proof Explorer

Theorem dalem3

Description: Lemma for dalemdnee . (Contributed by NM, 10-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 )
dalem3.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem3.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem3.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem3.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem3.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
dalem3.e ⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) )
Assertion dalem3 ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → 𝐷 ≠ 𝐸 )

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 dalem3.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dalem3.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
7 dalem3.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
8 dalem3.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
9 dalem3.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
10 dalem3.e ⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) )
11 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
12 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
13 1 dalemqea ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
14 1 dalemrea ⊢ ( 𝜑 → 𝑅 ∈ 𝐴 )
15 1 dalemyeo ⊢ ( 𝜑 → 𝑌 ∈ 𝑂 )
16 2 3 4 6 7 lplnric ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑂 ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
17 11 12 13 14 15 16 syl131anc ⊢ ( 𝜑 → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
18 17 adantr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
19 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
20 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
21 20 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
22 11 13 14 21 syl3anc ⊢ ( 𝜑 → ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
23 1 3 4 dalemtjueb ⊢ ( 𝜑 → ( 𝑇 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
24 20 2 5 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑇 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) ) ≤ ( 𝑄 ∨ 𝑅 ) )
25 19 22 23 24 syl3anc ⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) ) ≤ ( 𝑄 ∨ 𝑅 ) )
26 10 25 eqbrtrid ⊢ ( 𝜑 → 𝐸 ≤ ( 𝑄 ∨ 𝑅 ) )
27 breq1 ⊢ ( 𝐷 = 𝐸 → ( 𝐷 ≤ ( 𝑄 ∨ 𝑅 ) ↔ 𝐸 ≤ ( 𝑄 ∨ 𝑅 ) ) )
28 26 27 syl5ibrcom ⊢ ( 𝜑 → ( 𝐷 = 𝐸 → 𝐷 ≤ ( 𝑄 ∨ 𝑅 ) ) )
29 28 adantr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → ( 𝐷 = 𝐸 → 𝐷 ≤ ( 𝑄 ∨ 𝑅 ) ) )
30 11 adantr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → 𝐾 ∈ HL )
31 1 2 3 4 5 6 7 8 9 dalemdea ⊢ ( 𝜑 → 𝐷 ∈ 𝐴 )
32 31 adantr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → 𝐷 ∈ 𝐴 )
33 14 adantr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → 𝑅 ∈ 𝐴 )
34 13 adantr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → 𝑄 ∈ 𝐴 )
35 simpr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → 𝐷 ≠ 𝑄 )
36 2 3 4 hlatexch1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐷 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝐷 ≠ 𝑄 ) → ( 𝐷 ≤ ( 𝑄 ∨ 𝑅 ) → 𝑅 ≤ ( 𝑄 ∨ 𝐷 ) ) )
37 30 32 33 34 35 36 syl131anc ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → ( 𝐷 ≤ ( 𝑄 ∨ 𝑅 ) → 𝑅 ≤ ( 𝑄 ∨ 𝐷 ) ) )
38 2 3 4 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) )
39 11 12 13 38 syl3anc ⊢ ( 𝜑 → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) )
40 1 3 4 dalempjqeb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
41 1 3 4 dalemsjteb ⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
42 20 2 5 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )
43 19 40 41 42 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )
44 9 43 eqbrtrid ⊢ ( 𝜑 → 𝐷 ≤ ( 𝑃 ∨ 𝑄 ) )
45 1 4 dalemqeb ⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
46 20 4 atbase ⊢ ( 𝐷 ∈ 𝐴 → 𝐷 ∈ ( Base ‘ 𝐾 ) )
47 31 46 syl ⊢ ( 𝜑 → 𝐷 ∈ ( Base ‘ 𝐾 ) )
48 20 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝐷 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐷 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑄 ∨ 𝐷 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
49 19 45 47 40 48 syl13anc ⊢ ( 𝜑 → ( ( 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐷 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑄 ∨ 𝐷 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
50 39 44 49 mpbi2and ⊢ ( 𝜑 → ( 𝑄 ∨ 𝐷 ) ≤ ( 𝑃 ∨ 𝑄 ) )
51 1 4 dalemreb ⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
52 20 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( 𝑄 ∨ 𝐷 ) ∈ ( Base ‘ 𝐾 ) )
53 11 13 31 52 syl3anc ⊢ ( 𝜑 → ( 𝑄 ∨ 𝐷 ) ∈ ( Base ‘ 𝐾 ) )
54 20 2 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝐷 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑅 ≤ ( 𝑄 ∨ 𝐷 ) ∧ ( 𝑄 ∨ 𝐷 ) ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
55 19 51 53 40 54 syl13anc ⊢ ( 𝜑 → ( ( 𝑅 ≤ ( 𝑄 ∨ 𝐷 ) ∧ ( 𝑄 ∨ 𝐷 ) ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
56 50 55 mpan2d ⊢ ( 𝜑 → ( 𝑅 ≤ ( 𝑄 ∨ 𝐷 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
57 56 adantr ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝐷 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
58 29 37 57 3syld ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → ( 𝐷 = 𝐸 → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
59 58 necon3bd ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) → 𝐷 ≠ 𝐸 ) )
60 18 59 mpd ⊢ ( ( 𝜑 ∧ 𝐷 ≠ 𝑄 ) → 𝐷 ≠ 𝐸 )