Step |
Hyp |
Ref |
Expression |
1 |
|
pmaplub.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pmaplub.u |
⊢ 𝑈 = ( lub ‘ 𝐾 ) |
3 |
|
pmaplub.m |
⊢ 𝑀 = ( pmap ‘ 𝐾 ) |
4 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
5 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
6 |
1 4 5 3
|
pmapval |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) = { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) |
7 |
6
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑈 ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) ) |
8 |
|
hlomcmat |
⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ) |
9 |
1 4 2 5
|
atlatmstc |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) = 𝑋 ) |
10 |
8 9
|
sylan |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) = 𝑋 ) |
11 |
7 10
|
eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ ( 𝑀 ‘ 𝑋 ) ) = 𝑋 ) |