Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnco.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
ltrnco.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
4 |
|
eqid |
⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
1 4 2
|
ltrnldil |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
6 |
5
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
7 |
1 4 2
|
ltrnldil |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → 𝐺 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
8 |
7
|
3adant2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → 𝐺 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
9 |
1 4
|
ldilco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝐺 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
10 |
3 6 8 9
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
11 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
simp2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑝 ∈ ( Atoms ‘ 𝐾 ) ) |
13 |
|
simp3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) |
14 |
12 13
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) |
15 |
|
simp2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) |
16 |
|
simp3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) |
17 |
15 16
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) |
18 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐹 ∈ 𝑇 ) |
19 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐺 ∈ 𝑇 ) |
20 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
21 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
22 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
23 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
24 |
20 21 22 23 1 2
|
cdlemg41 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ∧ ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
25 |
11 14 17 18 19 24
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
26 |
25
|
3exp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) → ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
27 |
26
|
ralrimivv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
28 |
20 21 22 23 1 4 2
|
isltrn |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) |
29 |
28
|
3ad2ant1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) |
30 |
10 27 29
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 ) |