Metamath Proof Explorer

Theorem cdleme25b

Description: Transform cdleme24 . TODO get rid of $d's on U , N (Contributed by NM, 1-Jan-2013)

Ref Expression
Hypotheses cdleme24.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme24.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme24.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme24.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme24.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme24.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme24.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme24.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
cdleme24.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) )
Assertion cdleme25b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )

Proof

Step Hyp Ref Expression
1 cdleme24.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme24.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme24.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme24.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme24.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme24.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme24.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme24.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9 cdleme24.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) )
10 1 2 3 4 5 6 7 8 9 cdleme25a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑁 ∈ 𝐵 ) )
11 eqid ⊢ ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
12 eqid ⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
13 1 2 3 4 5 6 7 8 9 11 12 cdleme24 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) )
14 breq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ≤ 𝑊 ↔ 𝑡 ≤ 𝑊 ) )
15 14 notbid ⊢ ( 𝑠 = 𝑡 → ( ¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑡 ≤ 𝑊 ) )
16 breq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
17 16 notbid ⊢ ( 𝑠 = 𝑡 → ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
18 15 17 anbi12d ⊢ ( 𝑠 = 𝑡 → ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
19 oveq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ∨ 𝑈 ) = ( 𝑡 ∨ 𝑈 ) )
20 oveq2 ⊢ ( 𝑠 = 𝑡 → ( 𝑃 ∨ 𝑠 ) = ( 𝑃 ∨ 𝑡 ) )
21 20 oveq1d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) )
22 21 oveq2d ⊢ ( 𝑠 = 𝑡 → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
23 19 22 oveq12d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) )
24 8 23 eqtrid ⊢ ( 𝑠 = 𝑡 → 𝐹 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) )
25 oveq2 ⊢ ( 𝑠 = 𝑡 → ( 𝑅 ∨ 𝑠 ) = ( 𝑅 ∨ 𝑡 ) )
26 25 oveq1d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) = ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) )
27 24 26 oveq12d ⊢ ( 𝑠 = 𝑡 → ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) = ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
28 27 oveq2d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) )
29 9 28 eqtrid ⊢ ( 𝑠 = 𝑡 → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) )
30 18 29 reusv3 ⊢ ( ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑁 ∈ 𝐵 ) → ( ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ↔ ∃ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) )
31 30 biimpd ⊢ ( ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑁 ∈ 𝐵 ) → ( ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) → ∃ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) )
32 10 13 31 sylc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )