Step |
Hyp |
Ref |
Expression |
1 |
|
pmap11.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pmap11.m |
⊢ 𝑀 = ( pmap ‘ 𝐾 ) |
3 |
|
eqss |
⊢ ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ) |
4 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
5 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
6 |
1 5
|
latasymb |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) ) |
7 |
4 6
|
syl3an1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) ) |
8 |
1 5 2
|
pmaple |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ↔ ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ) ) |
9 |
1 5 2
|
pmaple |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ) |
10 |
9
|
3com23 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ) |
11 |
8 10
|
anbi12d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ) ) |
12 |
7 11
|
bitr3d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = 𝑌 ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ) ) |
13 |
3 12
|
bitr4id |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |