Step |
Hyp |
Ref |
Expression |
1 |
|
pmapj2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pmapj2.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
pmapj2.m |
⊢ 𝑀 = ( pmap ‘ 𝐾 ) |
4 |
|
pmapj2.p |
⊢ + = ( +𝑃 ‘ 𝐾 ) |
5 |
|
pmapj2.o |
⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 ) |
6 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ HL ) |
7 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat ) |
9 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
10 |
9
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ OP ) |
11 |
|
simp2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
12 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
13 |
1 12
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
14 |
10 11 13
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
15 |
|
simp3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
16 |
1 12
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ 𝐵 ) |
17 |
10 15 16
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ 𝐵 ) |
18 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
19 |
1 18
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ∈ 𝐵 ) |
20 |
8 14 17 19
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ∈ 𝐵 ) |
21 |
1 12 3 5
|
polpmapN |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) ) |
22 |
6 20 21
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) ) |
23 |
1 12 3 5
|
polpmapN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) |
24 |
23
|
3adant3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) |
25 |
1 12 3 5
|
polpmapN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ 𝑌 ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
26 |
25
|
3adant2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ 𝑌 ) ) = ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
27 |
24 26
|
ineq12d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) ∩ ( ⊥ ‘ ( 𝑀 ‘ 𝑌 ) ) ) = ( ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ∩ ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) |
28 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
29 |
1 28 3
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
30 |
29
|
3adant3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
31 |
1 28 3
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
32 |
31
|
3adant2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
33 |
28 4 5
|
poldmj1N |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑀 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( ⊥ ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) = ( ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) ∩ ( ⊥ ‘ ( 𝑀 ‘ 𝑌 ) ) ) ) |
34 |
6 30 32 33
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) = ( ( ⊥ ‘ ( 𝑀 ‘ 𝑋 ) ) ∩ ( ⊥ ‘ ( 𝑀 ‘ 𝑌 ) ) ) ) |
35 |
1 18 28 3
|
pmapmeet |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑀 ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ∩ ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) |
36 |
6 14 17 35
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ∩ ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) |
37 |
27 34 36
|
3eqtr4rd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( ⊥ ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) ) |
38 |
37
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑀 ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) ) ) |
39 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
40 |
1 2 18 12
|
oldmm4 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( 𝑋 ∨ 𝑌 ) ) |
41 |
39 40
|
syl3an1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( 𝑋 ∨ 𝑌 ) ) |
42 |
41
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) = ( 𝑀 ‘ ( 𝑋 ∨ 𝑌 ) ) ) |
43 |
22 38 42
|
3eqtr3rd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ⊥ ‘ ( ⊥ ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) ) ) |