| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lautm.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
| 2 |
|
lautm.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
| 3 |
|
lautm.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
|
latmcl |
⊢ ( ( 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
|
latmcl |
⊢ ( ( 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 4 2
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 ) |
| 21 |
20
|
3adant3r1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 ) |
| 22 |
1 4 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ) ) |
| 23 |
7 9 12 22
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ) ) |
| 24 |
21 23
|
mpbid |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ) |
| 25 |
1 4 2
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑌 ) |
| 26 |
25
|
3adant3r1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑌 ) |
| 27 |
1 4 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑌 ↔ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) ) |
| 28 |
7 9 15 27
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑌 ↔ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) ) |
| 29 |
26 28
|
mpbid |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) |
| 30 |
1 4 2
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) ↔ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 31 |
5 11 14 17 30
|
syl13anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) ↔ ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 32 |
24 29 31
|
mpbi2and |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ( le ‘ 𝐾 ) ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) |
| 33 |
1 3
|
laut1o |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
| 34 |
33
|
3ad2antr1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
| 35 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ) = ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) |
| 36 |
34 19 35
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ) = ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) |
| 37 |
1 4 2
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ) |
| 38 |
5 14 17 37
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ) |
| 39 |
1 4 3
|
lautcnvle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ↔ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
| 40 |
7 19 14 39
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑋 ) ↔ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
| 41 |
38 40
|
mpbid |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
| 42 |
|
f1ocnvfv1 |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑋 ) ) = 𝑋 ) |
| 43 |
34 12 42
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑋 ) ) = 𝑋 ) |
| 44 |
41 43
|
breqtrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) 𝑋 ) |
| 45 |
1 4 2
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) |
| 46 |
5 14 17 45
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) |
| 47 |
1 4 3
|
lautcnvle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ↔ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 48 |
7 19 17 47
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ↔ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑌 ) ) ) ) |
| 49 |
46 48
|
mpbid |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑌 ) ) ) |
| 50 |
|
f1ocnvfv1 |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑌 ) ) = 𝑌 ) |
| 51 |
34 15 50
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑌 ) ) = 𝑌 ) |
| 52 |
49 51
|
breqtrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) 𝑌 ) |
| 53 |
|
f1ocnvdm |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ) |
| 54 |
34 19 53
|
syl2anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ) |
| 55 |
1 4 2
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) 𝑋 ∧ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) 𝑌 ) ↔ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑌 ) ) ) |
| 56 |
5 54 12 15 55
|
syl13anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) 𝑋 ∧ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) 𝑌 ) ↔ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑌 ) ) ) |
| 57 |
44 52 56
|
mpbi2and |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑌 ) ) |
| 58 |
1 4 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ 𝐼 ) ∧ ( ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) ) → ( ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑌 ) ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ) |
| 59 |
7 54 9 58
|
syl12anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑌 ) ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ) |
| 60 |
57 59
|
mpbid |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ) |
| 61 |
36 60
|
eqbrtrrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ( le ‘ 𝐾 ) ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) ) |
| 62 |
1 4 5 11 19 32 61
|
latasymd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑌 ) ) ) |