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 |
|
cdleme27.g |
⊢ 𝐺 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
14 |
|
cdleme27.o |
⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) ) |
15 |
|
cdleme27.e |
⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) ) |
16 |
|
cdleme27.y |
⊢ 𝑌 = if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 ) |
17 |
|
simp211 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
18 |
|
simp221 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
19 |
|
simp222 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
20 |
|
simp213 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) |
21 |
|
simp223 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) |
22 |
|
simp23r |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
23 |
|
simp212 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 ) |
24 |
|
simp1l |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) |
25 |
|
simp1r |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) |
26 |
23 24 25
|
3jca |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
27 |
|
simp3 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) |
28 |
1 2 3 4 5 6 7 9 10 14 11 15
|
cdleme26ee |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) |
29 |
17 18 19 20 21 22 26 27 28
|
syl332anc |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) |
30 |
29
|
3expia |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) ) |
31 |
|
simp1r |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) |
32 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐾 ∈ HL ) |
33 |
32
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ HL ) |
34 |
|
simp13l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑠 ∈ 𝐴 ) |
35 |
34
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ∈ 𝐴 ) |
36 |
|
simp23l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑡 ∈ 𝐴 ) |
37 |
36
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑡 ∈ 𝐴 ) |
38 |
|
simp3ll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑠 ≠ 𝑡 ) |
39 |
38
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≠ 𝑡 ) |
40 |
35 37 39
|
3jca |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴 ∧ 𝑠 ≠ 𝑡 ) ) |
41 |
|
simp21l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑃 ∈ 𝐴 ) |
42 |
41
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ 𝐴 ) |
43 |
|
simp22l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑄 ∈ 𝐴 ) |
44 |
43
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ∈ 𝐴 ) |
45 |
|
simp212 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 ) |
46 |
|
simp3rl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑉 ∈ 𝐴 ) |
47 |
46
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑉 ∈ 𝐴 ) |
48 |
|
simp3 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) |
49 |
|
simp3lr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) |
50 |
49
|
3ad2ant2 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) |
51 |
|
simp1l |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) |
52 |
48 50 51
|
3jca |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
53 |
2 3 4 5 6
|
cdleme22b |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴 ∧ 𝑠 ≠ 𝑡 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑉 ∈ 𝐴 ∧ ( ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) |
54 |
33 40 42 44 45 47 52 53
|
syl232anc |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) |
55 |
31 54
|
pm2.21dd |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) |
56 |
55
|
3expia |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) ) |
57 |
30 56
|
pm2.61dne |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) |
58 |
|
iftrue |
⊢ ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) = 𝐷 ) |
59 |
12 58
|
syl5eq |
⊢ ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → 𝐶 = 𝐷 ) |
60 |
59
|
ad2antrr |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐷 ) |
61 |
|
iftrue |
⊢ ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 ) = 𝐸 ) |
62 |
16 61
|
syl5eq |
⊢ ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → 𝑌 = 𝐸 ) |
63 |
62
|
oveq1d |
⊢ ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐸 ∨ 𝑉 ) ) |
64 |
63
|
ad2antlr |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐸 ∨ 𝑉 ) ) |
65 |
57 60 64
|
3brtr4d |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) |
66 |
65
|
ex |
⊢ ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) ) |
67 |
|
simpr11 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
68 |
|
simpr12 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑃 ≠ 𝑄 ) |
69 |
|
simpll |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) |
70 |
68 69
|
jca |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
71 |
|
simpr23 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) |
72 |
|
simpr21 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
73 |
|
simpr22 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
74 |
|
simpr13 |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) |
75 |
|
simplr |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) |
76 |
|
simpr3l |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ) |
77 |
|
simpr3r |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
78 |
|
eqid |
⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
79 |
|
eqid |
⊢ ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) |
80 |
9 10 13 78 11 79
|
cdleme25cv |
⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) |
81 |
1 2 3 4 5 6 7 13 78 80
|
cdleme26f |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐷 ≤ ( 𝐺 ∨ 𝑉 ) ) |
82 |
67 70 71 72 73 74 75 76 77 81
|
syl333anc |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐷 ≤ ( 𝐺 ∨ 𝑉 ) ) |
83 |
59
|
ad2antrr |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐷 ) |
84 |
|
iffalse |
⊢ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 ) = 𝐺 ) |
85 |
16 84
|
syl5eq |
⊢ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → 𝑌 = 𝐺 ) |
86 |
85
|
oveq1d |
⊢ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐺 ∨ 𝑉 ) ) |
87 |
86
|
ad2antlr |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐺 ∨ 𝑉 ) ) |
88 |
82 83 87
|
3brtr4d |
⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) |
89 |
88
|
ex |
⊢ ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) ) |
90 |
|
simpr11 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
91 |
|
simpr12 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑃 ≠ 𝑄 ) |
92 |
|
simplr |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) |
93 |
91 92
|
jca |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
94 |
|
simpr13 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) |
95 |
|
simpr21 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
96 |
|
simpr22 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
97 |
|
simpr23 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) |
98 |
|
simpll |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) |
99 |
|
simpr3l |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ) |
100 |
|
simpr3r |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
101 |
|
eqid |
⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
102 |
|
eqid |
⊢ ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) ) ) |
103 |
9 14 8 101 15 102
|
cdleme25cv |
⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) ) ) |
104 |
1 2 3 4 5 6 7 8 101 103
|
cdleme26f2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐹 ≤ ( 𝐸 ∨ 𝑉 ) ) |
105 |
90 93 94 95 96 97 98 99 100 104
|
syl333anc |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐹 ≤ ( 𝐸 ∨ 𝑉 ) ) |
106 |
|
iffalse |
⊢ ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) = 𝐹 ) |
107 |
12 106
|
syl5eq |
⊢ ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → 𝐶 = 𝐹 ) |
108 |
107
|
ad2antrr |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐹 ) |
109 |
63
|
ad2antlr |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐸 ∨ 𝑉 ) ) |
110 |
105 108 109
|
3brtr4d |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) |
111 |
110
|
ex |
⊢ ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) ) |
112 |
|
simpr11 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
113 |
|
simpr23 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) |
114 |
|
simplr |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) |
115 |
|
simpll |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) |
116 |
|
simpr12 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑃 ≠ 𝑄 ) |
117 |
114 115 116
|
3jca |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) |
118 |
|
simpr21 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
119 |
|
simpr22 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
120 |
|
simpr13 |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) |
121 |
|
simpr3l |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ) |
122 |
|
simpr3r |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
123 |
2 3 4 5 6 7 8 13
|
cdleme22g |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐹 ≤ ( 𝐺 ∨ 𝑉 ) ) |
124 |
112 113 117 118 119 120 121 122 123
|
syl323anc |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐹 ≤ ( 𝐺 ∨ 𝑉 ) ) |
125 |
107
|
ad2antrr |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐹 ) |
126 |
86
|
ad2antlr |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐺 ∨ 𝑉 ) ) |
127 |
124 125 126
|
3brtr4d |
⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) |
128 |
127
|
ex |
⊢ ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) ) |
129 |
66 89 111 128
|
4cases |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) |