Step |
Hyp |
Ref |
Expression |
1 |
|
pmapmeet.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pmapmeet.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
3 |
|
pmapmeet.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
pmapmeet.p |
⊢ 𝑃 = ( pmap ‘ 𝐾 ) |
5 |
|
eqid |
⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 ) |
6 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ HL ) |
7 |
|
simp2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
8 |
|
simp3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
9 |
5 2 6 7 8
|
meetval |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) = ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) |
10 |
9
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( 𝑃 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) ) |
11 |
|
prssi |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → { 𝑋 , 𝑌 } ⊆ 𝐵 ) |
12 |
11
|
3adant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → { 𝑋 , 𝑌 } ⊆ 𝐵 ) |
13 |
|
prnzg |
⊢ ( 𝑋 ∈ 𝐵 → { 𝑋 , 𝑌 } ≠ ∅ ) |
14 |
13
|
3ad2ant2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → { 𝑋 , 𝑌 } ≠ ∅ ) |
15 |
1 5 4
|
pmapglb |
⊢ ( ( 𝐾 ∈ HL ∧ { 𝑋 , 𝑌 } ⊆ 𝐵 ∧ { 𝑋 , 𝑌 } ≠ ∅ ) → ( 𝑃 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) = ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) ) |
16 |
6 12 14 15
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) = ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) ) |
17 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ 𝑋 ) ) |
18 |
|
fveq2 |
⊢ ( 𝑥 = 𝑌 → ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ 𝑌 ) ) |
19 |
17 18
|
iinxprg |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑋 ) ∩ ( 𝑃 ‘ 𝑌 ) ) ) |
20 |
19
|
3adant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑋 ) ∩ ( 𝑃 ‘ 𝑌 ) ) ) |
21 |
10 16 20
|
3eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝑃 ‘ 𝑋 ) ∩ ( 𝑃 ‘ 𝑌 ) ) ) |