Step |
Hyp |
Ref |
Expression |
1 |
|
djaj.k |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
djaj.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
djaj.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
djaj.j |
⊢ 𝐽 = ( ( vA ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
6 |
5
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝐾 ∈ Lat ) |
7 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
8 |
7
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝐾 ∈ OP ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
10 |
9 2 3
|
diadmclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
11 |
10
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
12 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
13 |
9 12
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
14 |
8 11 13
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
15 |
9 2
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
16 |
15
|
ad2antlr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
17 |
9 12
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
8 16 17
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
9 1
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
6 14 18 19
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
21 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
22 |
9 21
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
6 20 16 22
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
24 |
9 2 3
|
diadmclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ dom 𝐼 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
25 |
24
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
26 |
9 12
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) |
27 |
8 25 26
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) |
28 |
9 1
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
29 |
6 27 18 28
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
30 |
9 21
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
31 |
6 29 16 30
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
32 |
9 21
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
33 |
6 23 31 32
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
34 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
35 |
9 34 21
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( le ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
36 |
6 23 31 35
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( le ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
37 |
9 34 21
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) |
38 |
6 29 16 37
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) |
39 |
9 34 6 33 31 16 36 38
|
lattrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( le ‘ 𝐾 ) 𝑊 ) |
40 |
9 34 2 3
|
diaeldm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ dom 𝐼 ↔ ( ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
41 |
40
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ dom 𝐼 ↔ ( ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
42 |
33 39 41
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ dom 𝐼 ) |
43 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
44 |
|
eqid |
⊢ ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) |
45 |
1 21 12 2 43 3 44
|
diaocN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) |
46 |
42 45
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) |
47 |
|
hloml |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OML ) |
48 |
47
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝐾 ∈ OML ) |
49 |
9 1
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑋 ∨ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) |
50 |
6 11 25 49
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑋 ∨ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) |
51 |
34 2 3
|
diadmleN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ( le ‘ 𝐾 ) 𝑊 ) |
52 |
51
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝑋 ( le ‘ 𝐾 ) 𝑊 ) |
53 |
34 2 3
|
diadmleN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ dom 𝐼 ) → 𝑌 ( le ‘ 𝐾 ) 𝑊 ) |
54 |
53
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝑌 ( le ‘ 𝐾 ) 𝑊 ) |
55 |
9 34 1
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑊 ) ↔ ( 𝑋 ∨ 𝑌 ) ( le ‘ 𝐾 ) 𝑊 ) ) |
56 |
6 11 25 16 55
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑊 ) ↔ ( 𝑋 ∨ 𝑌 ) ( le ‘ 𝐾 ) 𝑊 ) ) |
57 |
52 54 56
|
mpbi2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑋 ∨ 𝑌 ) ( le ‘ 𝐾 ) 𝑊 ) |
58 |
9 34 1 21 12
|
omlspjN |
⊢ ( ( 𝐾 ∈ OML ∧ ( ( 𝑋 ∨ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑋 ∨ 𝑌 ) ( le ‘ 𝐾 ) 𝑊 ) → ( ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( 𝑋 ∨ 𝑌 ) ) |
59 |
48 50 16 57 58
|
syl121anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( 𝑋 ∨ 𝑌 ) ) |
60 |
9 1
|
latjidm |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) |
61 |
6 18 60
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) |
62 |
61
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
63 |
9 1
|
latjass |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑋 ∨ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
64 |
6 50 18 18 63
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
65 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
66 |
65
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝐾 ∈ OL ) |
67 |
9 1 21 12
|
oldmm2 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∨ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
68 |
66 50 16 67
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
69 |
9 1 21 12
|
oldmj1 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
70 |
66 11 25 69
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
71 |
9 34 21
|
latleeqm1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) = 𝑋 ) ) |
72 |
6 11 16 71
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) = 𝑋 ) ) |
73 |
52 72
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) = 𝑋 ) |
74 |
73
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) |
75 |
9 1 21 12
|
oldmm1 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
76 |
66 11 16 75
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
77 |
74 76
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
78 |
9 34 21
|
latleeqm1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑌 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) = 𝑌 ) ) |
79 |
6 25 16 78
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑌 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) = 𝑌 ) ) |
80 |
54 79
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) = 𝑌 ) |
81 |
80
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) |
82 |
9 1 21 12
|
oldmm1 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
83 |
66 25 16 82
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
84 |
81 83
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
85 |
77 84
|
oveq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
86 |
70 85
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
87 |
86
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
88 |
9 21
|
latmmdir |
⊢ ( ( 𝐾 ∈ OL ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
89 |
66 20 29 16 88
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
90 |
87 89
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
91 |
90
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∨ 𝑌 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
92 |
68 91
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
93 |
92
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
94 |
64 93
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
95 |
62 94
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
96 |
95
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
97 |
59 96
|
eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑋 ∨ 𝑌 ) = ( ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
98 |
97
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
99 |
|
simpl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
100 |
2 3
|
diaclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 ) |
101 |
100
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 ) |
102 |
2 43 3
|
diaelrnN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
103 |
101 102
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
104 |
2 3
|
diaclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑌 ) ∈ ran 𝐼 ) |
105 |
104
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ 𝑌 ) ∈ ran 𝐼 ) |
106 |
2 43 3
|
diaelrnN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐼 ‘ 𝑌 ) ∈ ran 𝐼 ) → ( 𝐼 ‘ 𝑌 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
107 |
105 106
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ 𝑌 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
108 |
2 43 3 44 4
|
djavalN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐼 ‘ 𝑋 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝐼 ‘ 𝑌 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( ( 𝐼 ‘ 𝑋 ) 𝐽 ( 𝐼 ‘ 𝑌 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑋 ) ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑌 ) ) ) ) ) |
109 |
99 103 107 108
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝐼 ‘ 𝑋 ) 𝐽 ( 𝐼 ‘ 𝑌 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑋 ) ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑌 ) ) ) ) ) |
110 |
9 34 21
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) |
111 |
6 20 16 110
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) |
112 |
9 34 2 3
|
diaeldm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
113 |
112
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
114 |
23 111 113
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ) |
115 |
9 34 2 3
|
diaeldm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
116 |
115
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
117 |
31 38 116
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ) |
118 |
21 2 3
|
diameetN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∩ ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
119 |
99 114 117 118
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∩ ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
120 |
1 21 12 2 43 3 44
|
diaocN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑋 ) ) ) |
121 |
120
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑋 ) ) ) |
122 |
1 21 12 2 43 3 44
|
diaocN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑌 ) ) ) |
123 |
122
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑌 ) ) ) |
124 |
121 123
|
ineq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∩ ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑋 ) ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑌 ) ) ) ) |
125 |
119 124
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑋 ) ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑌 ) ) ) ) |
126 |
125
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑋 ) ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ 𝑌 ) ) ) ) ) |
127 |
109 126
|
eqtr4d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( ( 𝐼 ‘ 𝑋 ) 𝐽 ( 𝐼 ‘ 𝑌 ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐼 ‘ ( ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( meet ‘ 𝐾 ) ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) |
128 |
46 98 127
|
3eqtr4d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) 𝐽 ( 𝐼 ‘ 𝑌 ) ) ) |