Step |
Hyp |
Ref |
Expression |
1 |
|
cmt2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cmt2.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
3 |
|
cmt2.c |
⊢ 𝐶 = ( cm ‘ 𝐾 ) |
4 |
|
omllat |
⊢ ( 𝐾 ∈ OML → 𝐾 ∈ Lat ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat ) |
6 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
7 |
1 6
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ) |
8 |
4 7
|
syl3an1 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ) |
9 |
|
simp2 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
10 |
|
omlop |
⊢ ( 𝐾 ∈ OML → 𝐾 ∈ OP ) |
11 |
10
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ OP ) |
12 |
|
simp3 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
13 |
1 2
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) |
14 |
11 12 13
|
syl2anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) |
15 |
1 6
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ∈ 𝐵 ) |
16 |
5 9 14 15
|
syl3anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ∈ 𝐵 ) |
17 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
18 |
1 17
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ∧ ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ∈ 𝐵 ) → ( ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ) = ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ) ) |
19 |
5 8 16 18
|
syl3anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ) = ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ) ) |
20 |
1 2
|
opococ |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 ) |
21 |
11 12 20
|
syl2anc |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 ) |
22 |
21
|
oveq2d |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) = ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ) |
23 |
22
|
oveq2d |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ) = ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ) ) |
24 |
19 23
|
eqtr4d |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ) = ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ) ) |
25 |
24
|
eqeq2d |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = ( ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ) ↔ 𝑋 = ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ) ) ) |
26 |
1 17 6 2 3
|
cmtvalN |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ 𝑋 = ( ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ) ) ) |
27 |
1 17 6 2 3
|
cmtvalN |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 𝐶 ( ⊥ ‘ 𝑌 ) ↔ 𝑋 = ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ) ) ) |
28 |
14 27
|
syld3an3 |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 ( ⊥ ‘ 𝑌 ) ↔ 𝑋 = ( ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑌 ) ) ( join ‘ 𝐾 ) ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) ) ) ) |
29 |
25 26 28
|
3bitr4d |
⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ 𝑋 𝐶 ( ⊥ ‘ 𝑌 ) ) ) |