Step |
Hyp |
Ref |
Expression |
1 |
|
pmapocj.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pmapocj.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
pmapocj.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
pmapocj.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
5 |
|
pmapocj.f |
⊢ 𝐹 = ( pmap ‘ 𝐾 ) |
6 |
|
pmapocj.p |
⊢ + = ( +𝑃 ‘ 𝐾 ) |
7 |
|
pmapocj.r |
⊢ 𝑁 = ( ⊥𝑃 ‘ 𝐾 ) |
8 |
1 2 5 6 7
|
pmapj2N |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( 𝑁 ‘ ( 𝑁 ‘ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) ) ) |
9 |
8
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) = ( 𝑁 ‘ ( 𝑁 ‘ ( 𝑁 ‘ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) ) ) ) |
10 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ HL ) |
11 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
12 |
1 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) |
13 |
11 12
|
syl3an1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) |
14 |
1 4 5 7
|
polpmapN |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) = ( 𝐹 ‘ ( ⊥ ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
15 |
10 13 14
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) = ( 𝐹 ‘ ( ⊥ ‘ ( 𝑋 ∨ 𝑌 ) ) ) ) |
16 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
17 |
1 16 5
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
18 |
17
|
3adant3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
19 |
1 16 5
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
20 |
19
|
3adant2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
21 |
16 6
|
paddssat |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
22 |
10 18 20 21
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
23 |
16 7
|
3polN |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( 𝑁 ‘ ( 𝑁 ‘ ( 𝑁 ‘ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) ) ) = ( 𝑁 ‘ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) ) |
24 |
10 22 23
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ ( 𝑁 ‘ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) ) ) = ( 𝑁 ‘ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) ) |
25 |
9 15 24
|
3eqtr3d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ ( ⊥ ‘ ( 𝑋 ∨ 𝑌 ) ) ) = ( 𝑁 ‘ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ) ) |