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Metamath Proof Explorer

Theorem 4atexlemc

Description: Lemma for 4atexlem7 . (Contributed by NM, 24-Nov-2012)

Ref Expression
Hypotheses 4thatlem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
4thatlem0.l ⊢ ≤ = ( le ‘ 𝐾 )
4thatlem0.j ⊢ ∨ = ( join ‘ 𝐾 )
4thatlem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
4thatlem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4thatlem0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4thatlem0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
4thatlem0.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
4thatlem0.c ⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
Assertion 4atexlemc ( 𝜑 → 𝐶 ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 4thatlem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
2 4thatlem0.l ⊢ ≤ = ( le ‘ 𝐾 )
3 4thatlem0.j ⊢ ∨ = ( join ‘ 𝐾 )
4 4thatlem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 4thatlem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 4thatlem0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 4thatlem0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 4thatlem0.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
9 4thatlem0.c ⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
10 1 4atexlemkl ⊢ ( 𝜑 → 𝐾 ∈ Lat )
11 1 3 5 4atexlemqtb ⊢ ( 𝜑 → ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
12 1 3 5 4atexlempsb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
13 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 4 latmcom ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) )
15 10 11 12 14 syl3anc ⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) )
16 9 15 eqtrid ⊢ ( 𝜑 → 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) )
17 1 4atexlemk ⊢ ( 𝜑 → 𝐾 ∈ HL )
18 1 4atexlemp ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
19 1 4atexlems ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
20 1 4atexlemq ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
21 1 4atexlemt ⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
22 1 2 3 5 4atexlempns ⊢ ( 𝜑 → 𝑃 ≠ 𝑆 )
23 1 2 3 4 5 6 7 8 4atexlemntlpq ⊢ ( 𝜑 → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )
24 2 3 5 atnlej2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑇 ≠ 𝑄 )
25 24 necomd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ≠ 𝑇 )
26 17 21 18 20 23 25 syl131anc ⊢ ( 𝜑 → 𝑄 ≠ 𝑇 )
27 1 4atexlempnq ⊢ ( 𝜑 → 𝑃 ≠ 𝑄 )
28 1 4atexlemnslpq ⊢ ( 𝜑 → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
29 2 3 5 4atlem0ae ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) )
30 17 18 20 19 27 28 29 syl132anc ⊢ ( 𝜑 → ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) )
31 13 5 atbase ⊢ ( 𝑇 ∈ 𝐴 → 𝑇 ∈ ( Base ‘ 𝐾 ) )
32 21 31 syl ⊢ ( 𝜑 → 𝑇 ∈ ( Base ‘ 𝐾 ) )
33 1 2 3 4 5 6 7 4atexlemu ⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
34 1 2 3 4 5 6 7 8 4atexlemv ⊢ ( 𝜑 → 𝑉 ∈ 𝐴 )
35 13 3 5 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ) → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
36 17 33 34 35 syl3anc ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
37 13 5 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
38 20 37 syl ⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
39 13 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
40 10 12 38 39 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
41 1 4atexlemkc ⊢ ( 𝜑 → 𝐾 ∈ CvLat )
42 1 2 3 4 5 6 7 8 4atexlemunv ⊢ ( 𝜑 → 𝑈 ≠ 𝑉 )
43 1 4atexlemutvt ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) )
44 5 2 3 cvlsupr4 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ∧ ( 𝑈 ≠ 𝑉 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) → 𝑇 ≤ ( 𝑈 ∨ 𝑉 ) )
45 41 33 34 21 42 43 44 syl132anc ⊢ ( 𝜑 → 𝑇 ≤ ( 𝑈 ∨ 𝑉 ) )
46 13 3 5 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
47 17 18 20 46 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
48 1 6 4atexlemwb ⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
49 13 2 4 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
50 10 47 48 49 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
51 7 50 eqbrtrid ⊢ ( 𝜑 → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) )
52 13 2 4 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) )
53 10 12 48 52 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) )
54 8 53 eqbrtrid ⊢ ( 𝜑 → 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) )
55 13 5 atbase ⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
56 33 55 syl ⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
57 13 5 atbase ⊢ ( 𝑉 ∈ 𝐴 → 𝑉 ∈ ( Base ‘ 𝐾 ) )
58 34 57 syl ⊢ ( 𝜑 → 𝑉 ∈ ( Base ‘ 𝐾 ) )
59 13 2 3 latjlej12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ) → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) ) )
60 10 56 47 58 12 59 syl122anc ⊢ ( 𝜑 → ( ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ) → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) ) )
61 51 54 60 mp2and ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) )
62 3 5 hlatjass ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) )
63 17 18 20 19 62 syl13anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) )
64 13 5 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
65 18 64 syl ⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
66 13 5 atbase ⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) )
67 19 66 syl ⊢ ( 𝜑 → 𝑆 ∈ ( Base ‘ 𝐾 ) )
68 13 3 latj32 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) )
69 10 65 38 67 68 syl13anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) )
70 13 3 latjjdi ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) )
71 10 65 38 67 70 syl13anc ⊢ ( 𝜑 → ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) )
72 63 69 71 3eqtr3rd ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) )
73 61 72 breqtrd ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) )
74 13 2 10 32 36 40 45 73 lattrd ⊢ ( 𝜑 → 𝑇 ≤ ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) )
75 2 3 4 5 2atmat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ∧ ( 𝑄 ≠ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝑇 ≤ ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) ∈ 𝐴 )
76 17 18 19 20 21 22 26 30 74 75 syl333anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) ∈ 𝐴 )
77 16 76 eqeltrd ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )