Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemef50.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemef50.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdlemef50.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdlemef50.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdlemef50.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdlemef50.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdlemef50.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdlemef50.d |
⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
9 |
|
cdlemefs50.e |
⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
10 |
|
cdlemef50.f |
⊢ 𝐹 = ( 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 ) ) |
11 |
|
cdlemef50.v |
⊢ 𝑉 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 ) |
12 |
|
cdlemef50.n |
⊢ 𝑁 = ( ( 𝑣 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑣 ) ∧ 𝑊 ) ) ) |
13 |
|
cdlemefs50.o |
⊢ 𝑂 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝑁 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) ) |
14 |
|
cdlemef50.g |
⊢ 𝐺 = ( 𝑎 ∈ 𝐵 ↦ if ( ( 𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊 ) , ( ℩ 𝑐 ∈ 𝐵 ∀ 𝑢 ∈ 𝐴 ( ( ¬ 𝑢 ≤ 𝑊 ∧ ( 𝑢 ∨ ( 𝑎 ∧ 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ≤ ( 𝑄 ∨ 𝑃 ) , ( ℩ 𝑏 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑄 ∨ 𝑃 ) ) → 𝑏 = 𝑂 ) ) , ⦋ 𝑢 / 𝑣 ⦌ 𝑁 ) ∨ ( 𝑎 ∧ 𝑊 ) ) ) ) , 𝑎 ) ) |
15 |
1 2 3 4 5 6 7 8 9 10
|
cdleme50f |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 : 𝐵 ⟶ 𝐵 ) |
16 |
15
|
frnd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ran 𝐹 ⊆ 𝐵 ) |
17 |
1 2 3 4 5 6 11 12 13 14
|
cdlemeg46fvcl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑒 ) ∈ 𝐵 ) |
18 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
cdleme48fgv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑒 ) ) = 𝑒 ) |
19 |
|
fveqeq2 |
⊢ ( 𝑑 = ( 𝐺 ‘ 𝑒 ) → ( ( 𝐹 ‘ 𝑑 ) = 𝑒 ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑒 ) ) = 𝑒 ) ) |
20 |
19
|
rspcev |
⊢ ( ( ( 𝐺 ‘ 𝑒 ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑒 ) ) = 𝑒 ) → ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 ) |
21 |
17 18 20
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 ) |
22 |
15
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → 𝐹 : 𝐵 ⟶ 𝐵 ) |
23 |
|
ffn |
⊢ ( 𝐹 : 𝐵 ⟶ 𝐵 → 𝐹 Fn 𝐵 ) |
24 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝐵 → ( 𝑒 ∈ ran 𝐹 ↔ ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 ) ) |
25 |
22 23 24
|
3syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → ( 𝑒 ∈ ran 𝐹 ↔ ∃ 𝑑 ∈ 𝐵 ( 𝐹 ‘ 𝑑 ) = 𝑒 ) ) |
26 |
21 25
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑒 ∈ 𝐵 ) → 𝑒 ∈ ran 𝐹 ) |
27 |
16 26
|
eqelssd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ran 𝐹 = 𝐵 ) |