| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lecmt.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
| 2 |
|
lecmt.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
| 3 |
|
lecmt.c |
⊢ 𝐶 = ( cm ‘ 𝐾 ) |
| 4 |
|
omllat |
⊢ ( 𝐾 ∈ OML → 𝐾 ∈ Lat ) |
| 5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat ) |
| 6 |
|
simp2 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
| 7 |
|
omlop |
⊢ ( 𝐾 ∈ OML → 𝐾 ∈ OP ) |
| 8 |
7
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ OP ) |
| 9 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
| 10 |
1 9
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
| 11 |
8 6 10
|
syl2anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
| 12 |
|
simp3 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
| 13 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
| 14 |
1 13
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ) |
| 15 |
5 11 12 14
|
syl3anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ) |
| 16 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
| 17 |
1 2 16
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑋 ) |
| 18 |
5 6 15 17
|
syl3anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑋 ) |
| 19 |
1 16
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ∈ 𝐵 ) |
| 20 |
5 6 15 19
|
syl3anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ∈ 𝐵 ) |
| 21 |
1 2
|
lattr |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑌 ) ) |
| 22 |
5 20 6 12 21
|
syl13anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑌 ) ) |
| 23 |
18 22
|
mpand |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑌 ) ) |
| 24 |
1 2 13 16 9 3
|
cmtbr4N |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝑋 ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑌 ) ) ≤ 𝑌 ) ) |
| 25 |
23 24
|
sylibrd |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → 𝑋 𝐶 𝑌 ) ) |