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Metamath Proof Explorer

Theorem dalemcea

Description: Lemma for dath . Frequently-used utility lemma. Here we show that C must be an atom. This is an assumption in most presentations of Desargues's theorem; instead, we assume only the C is a lattice element, in order to make later substitutions for C easier. (Contributed by NM, 23-Sep-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 dalemcea ( 𝜑 → 𝐶 ∈ 𝐴 )

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 dalemkeop ⊢ ( 𝜑 → 𝐾 ∈ OP )
8 1 4 dalemceb ⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) )
9 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
10 1 2 3 4 5 6 dalempjsen ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( LLines ‘ 𝐾 ) )
11 1 dalemqea ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
12 1 dalemtea ⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
13 1 2 3 4 5 6 dalemqnet ⊢ ( 𝜑 → 𝑄 ≠ 𝑇 )
14 eqid ⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 )
15 3 4 14 llni2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ∧ 𝑄 ≠ 𝑇 ) → ( 𝑄 ∨ 𝑇 ) ∈ ( LLines ‘ 𝐾 ) )
16 9 11 12 13 15 syl31anc ⊢ ( 𝜑 → ( 𝑄 ∨ 𝑇 ) ∈ ( LLines ‘ 𝐾 ) )
17 1 2 3 4 5 6 dalem1 ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ≠ ( 𝑄 ∨ 𝑇 ) )
18 1 dalem-clpjq ⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) )
19 1 3 4 dalempjqeb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
20 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
21 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
22 20 2 21 op0le ⊢ ( ( 𝐾 ∈ OP ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( 0. ‘ 𝐾 ) ≤ ( 𝑃 ∨ 𝑄 ) )
23 7 19 22 syl2anc ⊢ ( 𝜑 → ( 0. ‘ 𝐾 ) ≤ ( 𝑃 ∨ 𝑄 ) )
24 breq1 ⊢ ( 𝐶 = ( 0. ‘ 𝐾 ) → ( 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ↔ ( 0. ‘ 𝐾 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
25 23 24 syl5ibrcom ⊢ ( 𝜑 → ( 𝐶 = ( 0. ‘ 𝐾 ) → 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) )
26 25 necon3bd ⊢ ( 𝜑 → ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) → 𝐶 ≠ ( 0. ‘ 𝐾 ) ) )
27 18 26 mpd ⊢ ( 𝜑 → 𝐶 ≠ ( 0. ‘ 𝐾 ) )
28 eqid ⊢ ( lt ‘ 𝐾 ) = ( lt ‘ 𝐾 )
29 20 28 21 opltn0 ⊢ ( ( 𝐾 ∈ OP ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) → ( ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) 𝐶 ↔ 𝐶 ≠ ( 0. ‘ 𝐾 ) ) )
30 7 8 29 syl2anc ⊢ ( 𝜑 → ( ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) 𝐶 ↔ 𝐶 ≠ ( 0. ‘ 𝐾 ) ) )
31 27 30 mpbird ⊢ ( 𝜑 → ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) 𝐶 )
32 1 dalemclpjs ⊢ ( 𝜑 → 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) )
33 1 dalemclqjt ⊢ ( 𝜑 → 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) )
34 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
35 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
36 1 dalemsea ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
37 20 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
38 9 35 36 37 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
39 20 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
40 9 11 12 39 syl3anc ⊢ ( 𝜑 → ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
41 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
42 20 2 41 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ) ↔ 𝐶 ≤ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ) )
43 34 8 38 40 42 syl13anc ⊢ ( 𝜑 → ( ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ) ↔ 𝐶 ≤ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ) )
44 32 33 43 mpbi2and ⊢ ( 𝜑 → 𝐶 ≤ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) )
45 opposet ⊢ ( 𝐾 ∈ OP → 𝐾 ∈ Poset )
46 7 45 syl ⊢ ( 𝜑 → 𝐾 ∈ Poset )
47 20 21 op0cl ⊢ ( 𝐾 ∈ OP → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
48 7 47 syl ⊢ ( 𝜑 → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
49 20 41 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ∈ ( Base ‘ 𝐾 ) )
50 34 38 40 49 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ∈ ( Base ‘ 𝐾 ) )
51 20 2 28 pltletr ⊢ ( ( 𝐾 ∈ Poset ∧ ( ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) 𝐶 ∧ 𝐶 ≤ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ) → ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ) )
52 46 48 8 50 51 syl13anc ⊢ ( 𝜑 → ( ( ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) 𝐶 ∧ 𝐶 ≤ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ) → ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ) )
53 31 44 52 mp2and ⊢ ( 𝜑 → ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) )
54 20 28 21 opltn0 ⊢ ( ( 𝐾 ∈ OP ∧ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ↔ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ≠ ( 0. ‘ 𝐾 ) ) )
55 7 50 54 syl2anc ⊢ ( 𝜑 → ( ( 0. ‘ 𝐾 ) ( lt ‘ 𝐾 ) ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ↔ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ≠ ( 0. ‘ 𝐾 ) ) )
56 53 55 mpbid ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ≠ ( 0. ‘ 𝐾 ) )
57 41 21 4 14 2llnmat ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( LLines ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝑇 ) ∈ ( LLines ‘ 𝐾 ) ) ∧ ( ( 𝑃 ∨ 𝑆 ) ≠ ( 𝑄 ∨ 𝑇 ) ∧ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ≠ ( 0. ‘ 𝐾 ) ) ) → ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ∈ 𝐴 )
58 9 10 16 17 56 57 syl32anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ∈ 𝐴 )
59 20 2 21 4 leat2 ⊢ ( ( ( 𝐾 ∈ OP ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ∈ 𝐴 ) ∧ ( 𝐶 ≠ ( 0. ‘ 𝐾 ) ∧ 𝐶 ≤ ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) ) ) → 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) )
60 7 8 58 27 44 59 syl32anc ⊢ ( 𝜑 → 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑇 ) ) )
61 60 58 eqeltrd ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )