Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme32.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdleme32.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdleme32.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdleme32.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdleme32.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdleme32.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdleme32.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdleme32.c |
⊢ 𝐶 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
9 |
|
cdleme32.d |
⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
10 |
|
cdleme32.e |
⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
11 |
|
cdleme32.i |
⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) ) |
12 |
|
cdleme32.n |
⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 ) |
13 |
1 2 3 4 5 6 7 8 9 10 11 12
|
cdleme32snaw |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ) → ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 ∧ ¬ ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≤ 𝑊 ) ) |
14 |
13
|
simpld |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 ) |
15 |
1 5
|
atbase |
⊢ ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 ) |
16 |
14 15
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 ) |