Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg4.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdlemg4.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
cdlemg4.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
cdlemg4.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
cdlemg4.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) |
7 |
6
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) 𝑃 ) ) |
8 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → 𝐾 ∈ HL ) |
9 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
simp23 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → 𝐺 ∈ 𝑇 ) |
11 |
|
simp21 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
12 |
1 2 3 4
|
ltrnel |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) |
13 |
12
|
simpld |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) |
14 |
9 10 11 13
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) |
15 |
|
simp21l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → 𝑃 ∈ 𝐴 ) |
16 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
17 |
16 2
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) 𝑃 ) = ( 𝑃 ( join ‘ 𝐾 ) ( 𝐺 ‘ 𝑃 ) ) ) |
18 |
8 14 15 17
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) 𝑃 ) = ( 𝑃 ( join ‘ 𝐾 ) ( 𝐺 ‘ 𝑃 ) ) ) |
19 |
7 18
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( 𝑃 ( join ‘ 𝐾 ) ( 𝐺 ‘ 𝑃 ) ) ) |
20 |
19
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
21 |
|
simp22 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → 𝐹 ∈ 𝑇 ) |
22 |
9 10 11 12
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) |
23 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
24 |
1 16 23 2 3 4 5
|
trlval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
25 |
9 21 22 24
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
26 |
1 16 23 2 3 4 5
|
trlval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
27 |
9 10 11 26
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
28 |
20 25 27
|
3eqtr4d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ) |