| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lautj.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
| 2 |
|
lautj.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
| 3 |
|
lautj.i |
⊢ 𝐼 = ( LAut ‘ 𝐾 ) |
| 4 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
| 5 |
|
simpl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐾 ∈ Lat ) |
| 6 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 ∈ 𝐼 ) |
| 7 |
5 6
|
jca |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ) |
| 8 |
1 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) |
| 9 |
8
|
3adant3r1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) |
| 10 |
1 3
|
lautcl |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∈ 𝐵 ) |
| 11 |
7 9 10
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∈ 𝐵 ) |
| 12 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
| 13 |
1 3
|
lautcl |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ) |
| 14 |
7 12 13
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ) |
| 15 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
| 16 |
1 3
|
lautcl |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) |
| 17 |
7 15 16
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) |
| 18 |
1 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ) |
| 19 |
5 14 17 18
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ) |
| 20 |
1 3
|
laut1o |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
| 21 |
20
|
3ad2antr1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
| 22 |
|
f1ocnvfv1 |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) = ( 𝑋 ∨ 𝑌 ) ) |
| 23 |
21 9 22
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) = ( 𝑋 ∨ 𝑌 ) ) |
| 24 |
1 4 2
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |
| 25 |
5 14 17 24
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |
| 26 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) = ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |
| 27 |
21 19 26
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) = ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |
| 28 |
25 27
|
breqtrrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) |
| 29 |
|
f1ocnvdm |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ) |
| 30 |
21 19 29
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ) |
| 31 |
1 4 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ) ) → ( 𝑋 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ↔ ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) ) |
| 32 |
7 12 30 31
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ↔ ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) ) |
| 33 |
28 32
|
mpbird |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 34 |
1 4 2
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |
| 35 |
5 14 17 34
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |
| 36 |
35 27
|
breqtrrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) |
| 37 |
1 4 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ) ) → ( 𝑌 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ↔ ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) ) |
| 38 |
7 15 30 37
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑌 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ↔ ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) ) |
| 39 |
36 38
|
mpbird |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 40 |
1 4 2
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ∧ 𝑌 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ↔ ( 𝑋 ∨ 𝑌 ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) |
| 41 |
5 12 15 30 40
|
syl13anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ∧ 𝑌 ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ↔ ( 𝑋 ∨ 𝑌 ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) |
| 42 |
33 39 41
|
mpbi2and |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∨ 𝑌 ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 43 |
23 42
|
eqbrtrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 44 |
1 4 3
|
lautcnvle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∈ 𝐵 ∧ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ) ) → ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ↔ ( ◡ 𝐹 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) |
| 45 |
7 11 19 44
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ↔ ( ◡ 𝐹 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) ) ) |
| 46 |
43 45
|
mpbird |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |
| 47 |
1 4 2
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ) |
| 48 |
47
|
3adant3r1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ) |
| 49 |
1 4 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) ) → ( 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ↔ ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
| 50 |
7 12 9 49
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ↔ ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
| 51 |
48 50
|
mpbid |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) |
| 52 |
1 4 2
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ) |
| 53 |
52
|
3adant3r1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ) |
| 54 |
1 4 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) ) → ( 𝑌 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ↔ ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
| 55 |
7 15 9 54
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑌 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑌 ) ↔ ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
| 56 |
53 55
|
mpbid |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) |
| 57 |
1 4 2
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∧ ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ↔ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
| 58 |
5 14 17 11 57
|
syl13anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∧ ( 𝐹 ‘ 𝑌 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ↔ ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
| 59 |
51 56 58
|
mpbi2and |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) |
| 60 |
1 4 5 11 19 46 59
|
latasymd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) ) |