Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemg2.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdlemg2.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdlemg2.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdlemg2.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdlemg2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdlemg2.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
cdlemg2ex.u |
⊢ 𝑈 = ( ( 𝑝 ∨ 𝑞 ) ∧ 𝑊 ) |
9 |
|
cdlemg2ex.d |
⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
10 |
|
cdlemg2ex.e |
⊢ 𝐸 = ( ( 𝑝 ∨ 𝑞 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
11 |
|
cdlemg2ex.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) ) |
12 |
2 5 6 7
|
cdlemg1cex |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) ) |
13 |
|
simplll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝐾 ∈ HL ) |
14 |
|
simpllr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 ) |
15 |
|
simplrl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑝 ∈ 𝐴 ) |
16 |
|
simprl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ¬ 𝑝 ≤ 𝑊 ) |
17 |
|
simplrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑞 ∈ 𝐴 ) |
18 |
|
simprr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ¬ 𝑞 ≤ 𝑊 ) |
19 |
|
eqid |
⊢ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) |
20 |
1 2 3 4 5 6 8 9 10 11 7 19
|
cdlemg1b2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) = 𝐺 ) |
21 |
13 14 15 16 17 18 20
|
syl222anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) = 𝐺 ) |
22 |
21
|
eqeq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ↔ 𝐹 = 𝐺 ) ) |
23 |
22
|
pm5.32da |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) → ( ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = 𝐺 ) ) ) |
24 |
|
df-3an |
⊢ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ) |
25 |
|
df-3an |
⊢ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ↔ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝐹 = 𝐺 ) ) |
26 |
23 24 25
|
3bitr4g |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) → ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ) ) |
27 |
26
|
2rexbidva |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ) ) |
28 |
12 27
|
bitrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺 ) ) ) |