Step |
Hyp |
Ref |
Expression |
1 |
|
dih11.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dih11.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
dih11.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
eqss |
⊢ ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 𝑌 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ) ) |
5 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
6 |
1 5 2 3
|
dihord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 ( le ‘ 𝐾 ) 𝑌 ) ) |
7 |
1 5 2 3
|
dihord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ) |
8 |
7
|
3com23 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ) |
9 |
6 8
|
anbi12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ) ↔ ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ) ) |
10 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ HL ) |
11 |
10
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat ) |
12 |
1 5
|
latasymb |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) ) |
13 |
11 12
|
syld3an1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) ) |
14 |
9 13
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ 𝑋 ) ) ↔ 𝑋 = 𝑌 ) ) |
15 |
4 14
|
syl5bb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) = ( 𝐼 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |