Metamath Proof Explorer

Theorem cdleme29b

Description: Transform cdleme28 . (Compare cdleme25b .) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013)

Ref Expression
Hypotheses cdleme26.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme26.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme26.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme26.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme26.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme26.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme27.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
cdleme27.z ⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.d ⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
cdleme27.c ⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
Assertion cdleme29b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme26.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme26.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme26.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme26.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme26.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme27.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9 cdleme27.z ⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
10 cdleme27.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
11 cdleme27.d ⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
12 cdleme27.c ⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
13 1 2 3 4 5 6 7 8 9 10 11 12 cdleme29ex ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝐵 ) )
14 eqid ⊢ ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
15 eqid ⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) )
16 eqid ⊢ ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) )
17 eqid ⊢ if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) = if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) )
18 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 cdleme28 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) ) )
19 breq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ≤ 𝑊 ↔ 𝑡 ≤ 𝑊 ) )
20 19 notbid ⊢ ( 𝑠 = 𝑡 → ( ¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑡 ≤ 𝑊 ) )
21 oveq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) )
22 21 eqeq1d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ↔ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
23 20 22 anbi12d ⊢ ( 𝑠 = 𝑡 → ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ↔ ( ¬ 𝑡 ≤ 𝑊 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) )
24 12 oveq1i ⊢ ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) ∨ ( 𝑋 ∧ 𝑊 ) )
25 breq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
26 oveq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ∨ 𝑧 ) = ( 𝑡 ∨ 𝑧 ) )
27 26 oveq1d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) = ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) )
28 27 oveq2d ⊢ ( 𝑠 = 𝑡 → ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) = ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) )
29 28 oveq2d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) )
30 10 29 syl5eq ⊢ ( 𝑠 = 𝑡 → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) )
31 30 eqeq2d ⊢ ( 𝑠 = 𝑡 → ( 𝑢 = 𝑁 ↔ 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) )
32 31 imbi2d ⊢ ( 𝑠 = 𝑡 → ( ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ↔ ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) )
33 32 ralbidv ⊢ ( 𝑠 = 𝑡 → ( ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ↔ ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) )
34 33 riotabidv ⊢ ( 𝑠 = 𝑡 → ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) )
35 11 34 syl5eq ⊢ ( 𝑠 = 𝑡 → 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) )
36 oveq1 ⊢ ( 𝑠 = 𝑡 → ( 𝑠 ∨ 𝑈 ) = ( 𝑡 ∨ 𝑈 ) )
37 oveq2 ⊢ ( 𝑠 = 𝑡 → ( 𝑃 ∨ 𝑠 ) = ( 𝑃 ∨ 𝑡 ) )
38 37 oveq1d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) )
39 38 oveq2d ⊢ ( 𝑠 = 𝑡 → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
40 36 39 oveq12d ⊢ ( 𝑠 = 𝑡 → ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) )
41 8 40 syl5eq ⊢ ( 𝑠 = 𝑡 → 𝐹 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) )
42 25 35 41 ifbieq12d ⊢ ( 𝑠 = 𝑡 → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) = if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) )
43 42 oveq1d ⊢ ( 𝑠 = 𝑡 → ( if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
44 24 43 syl5eq ⊢ ( 𝑠 = 𝑡 → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
45 23 44 reusv3 ⊢ ( ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝐵 ) → ( ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ∃ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) )
46 45 biimpd ⊢ ( ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝐵 ) → ( ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) ) ) , ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ∃ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) )
47 13 18 46 sylc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) )