Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg12.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdlemg12.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdlemg12.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdlemg12.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdlemg12.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdlemg12.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
cdlemg12b.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
simp31 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → 𝐹 ∈ 𝑇 ) |
10 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
11 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
12 |
1 2 3 4 5 6
|
ltrnu |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ) |
13 |
8 9 10 11 12
|
syl211anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ) |
14 |
|
simp33 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝐺 ‘ 𝑃 ) = 𝑃 ) |
15 |
14
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( 𝐹 ‘ 𝑃 ) ) |
16 |
15
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) |
17 |
16
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) ) |
18 |
|
simp32 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → 𝐺 ∈ 𝑇 ) |
19 |
1 4 5 6
|
ltrnateq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) → ( 𝐺 ‘ 𝑄 ) = 𝑄 ) |
20 |
8 18 10 11 14 19
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝐺 ‘ 𝑄 ) = 𝑄 ) |
21 |
20
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = ( 𝐹 ‘ 𝑄 ) ) |
22 |
21
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) |
23 |
22
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ) |
24 |
13 17 23
|
3eqtr4d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) = 𝑃 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑊 ) ) |