Step |
Hyp |
Ref |
Expression |
1 |
|
pmapojoin.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pmapojoin.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
pmapojoin.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
pmapojoin.m |
⊢ 𝑀 = ( pmap ‘ 𝐾 ) |
5 |
|
pmapojoin.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
6 |
|
pmapojoin.p |
⊢ + = ( +𝑃 ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( ⊥𝑃 ‘ 𝐾 ) = ( ⊥𝑃 ‘ 𝐾 ) |
8 |
1 3 4 6 7
|
pmapj2N |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) ) ) |
9 |
8
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( 𝑀 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) ) ) |
10 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → 𝐾 ∈ HL ) |
11 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → 𝑋 ∈ 𝐵 ) |
12 |
|
eqid |
⊢ ( PSubCl ‘ 𝐾 ) = ( PSubCl ‘ 𝐾 ) |
13 |
1 4 12
|
pmapsubclN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ∈ ( PSubCl ‘ 𝐾 ) ) |
14 |
10 11 13
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( 𝑀 ‘ 𝑋 ) ∈ ( PSubCl ‘ 𝐾 ) ) |
15 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ∈ 𝐵 ) |
16 |
1 4 12
|
pmapsubclN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑌 ) ∈ ( PSubCl ‘ 𝐾 ) ) |
17 |
10 15 16
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( 𝑀 ‘ 𝑌 ) ∈ ( PSubCl ‘ 𝐾 ) ) |
18 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
19 |
18
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ OP ) |
20 |
|
simp3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
21 |
1 5
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) |
22 |
19 20 21
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) |
23 |
1 2 4
|
pmaple |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ↔ ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ ( ⊥ ‘ 𝑌 ) ) ) ) |
24 |
22 23
|
syld3an3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ↔ ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ ( ⊥ ‘ 𝑌 ) ) ) ) |
25 |
24
|
biimpa |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ ( ⊥ ‘ 𝑌 ) ) ) |
26 |
1 5 4 7
|
polpmapN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑌 ) ) = ( 𝑀 ‘ ( ⊥ ‘ 𝑌 ) ) ) |
27 |
10 15 26
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑌 ) ) = ( 𝑀 ‘ ( ⊥ ‘ 𝑌 ) ) ) |
28 |
25 27
|
sseqtrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( 𝑀 ‘ 𝑋 ) ⊆ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑌 ) ) ) |
29 |
6 7 12
|
osumclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑀 ‘ 𝑋 ) ∈ ( PSubCl ‘ 𝐾 ) ∧ ( 𝑀 ‘ 𝑌 ) ∈ ( PSubCl ‘ 𝐾 ) ) ∧ ( 𝑀 ‘ 𝑋 ) ⊆ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ 𝑌 ) ) ) → ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ∈ ( PSubCl ‘ 𝐾 ) ) |
30 |
10 14 17 28 29
|
syl31anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ∈ ( PSubCl ‘ 𝐾 ) ) |
31 |
7 12
|
psubcli2N |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ∈ ( PSubCl ‘ 𝐾 ) ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) ) = ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) |
32 |
10 30 31
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) ) = ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) |
33 |
9 32
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ ( ⊥ ‘ 𝑌 ) ) → ( 𝑀 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝑀 ‘ 𝑋 ) + ( 𝑀 ‘ 𝑌 ) ) ) |