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 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
15 |
|
simpl2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
16 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) |
17 |
|
simpl3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 ) |
18 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) |
19 |
1 2 3 4 5 6 7 9 10 11
|
cdleme25cl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐷 ∈ 𝐵 ) |
20 |
13 14 15 16 17 18 19
|
syl312anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐷 ∈ 𝐵 ) |
21 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝐾 ∈ HL ) |
22 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑊 ∈ 𝐻 ) |
23 |
|
simp2ll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ∈ 𝐴 ) |
24 |
|
simp2rl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑄 ∈ 𝐴 ) |
25 |
|
simp3ll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑠 ∈ 𝐴 ) |
26 |
2 3 4 5 6 7 8 1
|
cdleme1b |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ) ) → 𝐹 ∈ 𝐵 ) |
27 |
21 22 23 24 25 26
|
syl23anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝐹 ∈ 𝐵 ) |
28 |
27
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐹 ∈ 𝐵 ) |
29 |
20 28
|
ifclda |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) ∈ 𝐵 ) |
30 |
12 29
|
eqeltrid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝐶 ∈ 𝐵 ) |