Step |
Hyp |
Ref |
Expression |
1 |
|
djhj.k |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
djhj.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
djhj.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
djhj.j |
⊢ 𝐽 = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
djhj.w |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
djhj.x |
⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 ) |
7 |
|
djhj.y |
⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
9 |
8 2 3
|
dihcnvcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
10 |
5 6 9
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
11 |
8 2 3
|
dihcnvcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) |
12 |
5 7 11
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
8 1 2 3 4
|
djhlj |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ◡ 𝐼 ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑌 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑌 ) ) ) = ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) 𝐽 ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ) ) |
14 |
5 10 12 13
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑌 ) ) ) = ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) 𝐽 ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ) ) |
15 |
2 3
|
dihcnvid2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |
16 |
5 6 15
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |
17 |
2 3
|
dihcnvid2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) = 𝑌 ) |
18 |
5 7 17
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) = 𝑌 ) |
19 |
16 18
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) 𝐽 ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑌 ) ) ) = ( 𝑋 𝐽 𝑌 ) ) |
20 |
14 19
|
eqtr2d |
⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( 𝐼 ‘ ( ( ◡ 𝐼 ‘ 𝑋 ) ∨ ( ◡ 𝐼 ‘ 𝑌 ) ) ) ) |