Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemefr27.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemefr27.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdlemefr27.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdlemefr27.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdlemefr27.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdlemefr27.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdlemefr27.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdlemefr27.c |
⊢ 𝐶 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
9 |
|
cdlemefr27.n |
⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 ) |
10 |
|
simpr2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) |
11 |
10
|
iffalsed |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 ) = 𝐶 ) |
12 |
9 11
|
syl5eq |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑁 = 𝐶 ) |
13 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝐾 ∈ HL ) |
14 |
|
simpl1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑊 ∈ 𝐻 ) |
15 |
|
simpl2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ∈ 𝐴 ) |
16 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑄 ∈ 𝐴 ) |
17 |
|
simpr1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑠 ∈ 𝐴 ) |
18 |
2 3 4 5 6 7 8 1
|
cdleme1b |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ) ) → 𝐶 ∈ 𝐵 ) |
19 |
13 14 15 16 17 18
|
syl23anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝐶 ∈ 𝐵 ) |
20 |
12 19
|
eqeltrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑁 ∈ 𝐵 ) |