Step |
Hyp |
Ref |
Expression |
1 |
|
paddatcl.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
paddatcl.p |
⊢ + = ( +𝑃 ‘ 𝐾 ) |
3 |
|
paddatcl.c |
⊢ 𝐶 = ( PSubCl ‘ 𝐾 ) |
4 |
|
hlclat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ CLat ) |
6 |
1 3
|
psubclssatN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → 𝑋 ⊆ 𝐴 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
7 1
|
atssbase |
⊢ 𝐴 ⊆ ( Base ‘ 𝐾 ) |
9 |
6 8
|
sstrdi |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → 𝑋 ⊆ ( Base ‘ 𝐾 ) ) |
10 |
9
|
3adant3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → 𝑋 ⊆ ( Base ‘ 𝐾 ) ) |
11 |
|
eqid |
⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 ) |
12 |
7 11
|
clatlubcl |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑋 ⊆ ( Base ‘ 𝐾 ) ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
5 10 12
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
14 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
15 |
|
eqid |
⊢ ( pmap ‘ 𝐾 ) = ( pmap ‘ 𝐾 ) |
16 |
7 14 1 15 2
|
pmapjat1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) + ( ( pmap ‘ 𝐾 ) ‘ 𝑄 ) ) ) |
17 |
13 16
|
syld3an2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) + ( ( pmap ‘ 𝐾 ) ‘ 𝑄 ) ) ) |
18 |
11 15 3
|
pmapidclN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 ) |
19 |
18
|
3adant3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 ) |
20 |
1 15
|
pmapat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ 𝑄 ) = { 𝑄 } ) |
21 |
20
|
3adant2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ 𝑄 ) = { 𝑄 } ) |
22 |
19 21
|
oveq12d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) + ( ( pmap ‘ 𝐾 ) ‘ 𝑄 ) ) = ( 𝑋 + { 𝑄 } ) ) |
23 |
17 22
|
eqtr2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑋 + { 𝑄 } ) = ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ) ) |
24 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ HL ) |
25 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
26 |
25
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ Lat ) |
27 |
7 1
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
28 |
27
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
29 |
7 14
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
30 |
26 13 28 29
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
31 |
7 15 3
|
pmapsubclN |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ) ∈ 𝐶 ) |
32 |
24 30 31
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) 𝑄 ) ) ∈ 𝐶 ) |
33 |
23 32
|
eqeltrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑋 + { 𝑄 } ) ∈ 𝐶 ) |