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Metamath Proof Explorer

Theorem 4atexlemtlw

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 )
Assertion 4atexlemtlw ( 𝜑 → 𝑇 ≤ 𝑊 )

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 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10 1 4atexlemkl ⊢ ( 𝜑 → 𝐾 ∈ Lat )
11 1 4atexlemt ⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
12 9 5 atbase ⊢ ( 𝑇 ∈ 𝐴 → 𝑇 ∈ ( Base ‘ 𝐾 ) )
13 11 12 syl ⊢ ( 𝜑 → 𝑇 ∈ ( Base ‘ 𝐾 ) )
14 1 4atexlemk ⊢ ( 𝜑 → 𝐾 ∈ HL )
15 1 2 3 4 5 6 7 4atexlemu ⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
16 1 2 3 4 5 6 7 8 4atexlemv ⊢ ( 𝜑 → 𝑉 ∈ 𝐴 )
17 9 3 5 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ) → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
18 14 15 16 17 syl3anc ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
19 1 6 4atexlemwb ⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
20 1 4atexlemkc ⊢ ( 𝜑 → 𝐾 ∈ CvLat )
21 1 2 3 4 5 6 7 8 4atexlemunv ⊢ ( 𝜑 → 𝑈 ≠ 𝑉 )
22 1 4atexlemutvt ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) )
23 5 2 3 cvlsupr4 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ∧ ( 𝑈 ≠ 𝑉 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) → 𝑇 ≤ ( 𝑈 ∨ 𝑉 ) )
24 20 15 16 11 21 22 23 syl132anc ⊢ ( 𝜑 → 𝑇 ≤ ( 𝑈 ∨ 𝑉 ) )
25 1 4atexlemp ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
26 1 4atexlemq ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
27 9 3 5 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
28 14 25 26 27 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
29 9 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
30 10 28 19 29 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
31 7 30 eqbrtrid ⊢ ( 𝜑 → 𝑈 ≤ 𝑊 )
32 1 3 5 4atexlempsb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
33 9 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ 𝑊 )
34 10 32 19 33 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ 𝑊 )
35 8 34 eqbrtrid ⊢ ( 𝜑 → 𝑉 ≤ 𝑊 )
36 9 5 atbase ⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
37 15 36 syl ⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
38 9 5 atbase ⊢ ( 𝑉 ∈ 𝐴 → 𝑉 ∈ ( Base ‘ 𝐾 ) )
39 16 38 syl ⊢ ( 𝜑 → 𝑉 ∈ ( Base ‘ 𝐾 ) )
40 9 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊 ) ↔ ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 ) )
41 10 37 39 19 40 syl13anc ⊢ ( 𝜑 → ( ( 𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊 ) ↔ ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 ) )
42 31 35 41 mpbi2and ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ≤ 𝑊 )
43 9 2 10 13 18 19 24 42 lattrd ⊢ ( 𝜑 → 𝑇 ≤ 𝑊 )