Step |
Hyp |
Ref |
Expression |
1 |
|
dihjatc.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihjatc.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihjatc.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihjatc.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihjatc.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
dihjatc.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dihjatc.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
8 |
|
dihjatc.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dihjatc.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
dihjatc.x |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) |
11 |
|
dihjatc.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
12 |
9
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
13 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ OP ) |
15 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
16 |
1 15
|
op1cl |
⊢ ( 𝐾 ∈ OP → ( 1. ‘ 𝐾 ) ∈ 𝐵 ) |
17 |
14 16
|
syl |
⊢ ( 𝜑 → ( 1. ‘ 𝐾 ) ∈ 𝐵 ) |
18 |
10
|
simpld |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
19 |
11
|
simpld |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
20 |
1 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ 𝐵 ) |
22 |
1 2 15
|
ople1 |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑃 ∈ 𝐵 ) → 𝑃 ≤ ( 1. ‘ 𝐾 ) ) |
23 |
14 21 22
|
syl2anc |
⊢ ( 𝜑 → 𝑃 ≤ ( 1. ‘ 𝐾 ) ) |
24 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
25 |
12 24
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ OL ) |
26 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
27 |
1 26 15
|
olm12 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) = 𝑋 ) |
28 |
25 18 27
|
syl2anc |
⊢ ( 𝜑 → ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) = 𝑋 ) |
29 |
10
|
simprd |
⊢ ( 𝜑 → 𝑋 ≤ 𝑊 ) |
30 |
28 29
|
eqbrtrd |
⊢ ( 𝜑 → ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ≤ 𝑊 ) |
31 |
1 2 3 4 26 5 6 7 8
|
dihjatc3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 1. ‘ 𝐾 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ≤ ( 1. ‘ 𝐾 ) ∧ ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ∨ 𝑃 ) ) = ( ( 𝐼 ‘ ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) |
32 |
9 17 18 11 23 30 31
|
syl312anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ∨ 𝑃 ) ) = ( ( 𝐼 ‘ ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) |
33 |
28
|
fvoveq1d |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ∨ 𝑃 ) ) = ( 𝐼 ‘ ( 𝑋 ∨ 𝑃 ) ) ) |
34 |
28
|
fveq2d |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ) = ( 𝐼 ‘ 𝑋 ) ) |
35 |
34
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( ( 1. ‘ 𝐾 ) ( meet ‘ 𝐾 ) 𝑋 ) ) ⊕ ( 𝐼 ‘ 𝑃 ) ) = ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) |
36 |
32 33 35
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑋 ∨ 𝑃 ) ) = ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) |